<?xml version="1.0" encoding="UTF-8"?>

<!--XBRL Document Created with the Workiva Platform-->
<!--Copyright 2026 Workiva-->
<!--r:019e22b1-2149-75f3-bebd-7af366c94abe,g:e0114b51-0329-4e0e-8ea3-a180b9f6d30b-->
<link:linkbase xmlns:link="http://www.xbrl.org/2003/linkbase" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.xbrl.org/2003/linkbase http://www.xbrl.org/2003/xbrl-linkbase-2003-12-31.xsd">
  <link:roleRef roleURI="http://www.fgb.net/role/Cover" xlink:type="simple" xlink:href="fgbi-20260521.xsd#Cover"/>
  <link:presentationLink xlink:role="http://www.fgb.net/role/Cover" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_dei_CoverAbstract_019e22b1-2149-7f52-837e-1d17d84aa1a8" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_CoverAbstract"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_DocumentInformationTable_019e22b1-2149-728e-88e5-889f837c4567" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_DocumentInformationTable"/>
    <link:presentationArc order="1" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_CoverAbstract_019e22b1-2149-7f52-837e-1d17d84aa1a8" xlink:to="loc_dei_DocumentInformationTable_019e22b1-2149-728e-88e5-889f837c4567" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementClassOfStockAxis_019e22b1-2149-7157-892f-e623627e8599" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementClassOfStockAxis"/>
    <link:presentationArc order="1" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationTable_019e22b1-2149-728e-88e5-889f837c4567" xlink:to="loc_us-gaap_StatementClassOfStockAxis_019e22b1-2149-7157-892f-e623627e8599" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ClassOfStockDomain_019e22b1-2149-7aee-913c-0f4003916033" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ClassOfStockDomain"/>
    <link:presentationArc order="1" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_us-gaap_StatementClassOfStockAxis_019e22b1-2149-7157-892f-e623627e8599" xlink:to="loc_us-gaap_ClassOfStockDomain_019e22b1-2149-7aee-913c-0f4003916033" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockMember_019e22b1-2149-7e01-b75b-7acf5c8e1938" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockMember"/>
    <link:presentationArc order="1" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_us-gaap_ClassOfStockDomain_019e22b1-2149-7aee-913c-0f4003916033" xlink:to="loc_us-gaap_CommonStockMember_019e22b1-2149-7e01-b75b-7acf5c8e1938" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoncumulativePreferredStockMember_019e22b1-2149-7bd7-9841-c287ed98950d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoncumulativePreferredStockMember"/>
    <link:presentationArc order="2" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_us-gaap_ClassOfStockDomain_019e22b1-2149-7aee-913c-0f4003916033" xlink:to="loc_us-gaap_NoncumulativePreferredStockMember_019e22b1-2149-7bd7-9841-c287ed98950d" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_DocumentInformationLineItems"/>
    <link:presentationArc order="2" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationTable_019e22b1-2149-728e-88e5-889f837c4567" xlink:to="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_PreCommencementTenderOffer_019e22b1-2149-7dc2-8a69-55a3ca9aeb3d" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_PreCommencementTenderOffer"/>
    <link:presentationArc order="1" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_PreCommencementTenderOffer_019e22b1-2149-7dc2-8a69-55a3ca9aeb3d" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_PreCommencementIssuerTenderOffer_019e22b1-2149-7674-8440-388144a65fbe" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_PreCommencementIssuerTenderOffer"/>
    <link:presentationArc order="2" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_PreCommencementIssuerTenderOffer_019e22b1-2149-7674-8440-388144a65fbe" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityEmergingGrowthCompany_019e22b1-2149-79ee-ab41-295124c25fcd" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityEmergingGrowthCompany"/>
    <link:presentationArc order="3" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityEmergingGrowthCompany_019e22b1-2149-79ee-ab41-295124c25fcd" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_Security12bTitle_019e22b1-2149-7a9b-8e29-7d4aba50db8f" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_Security12bTitle"/>
    <link:presentationArc order="4" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_Security12bTitle_019e22b1-2149-7a9b-8e29-7d4aba50db8f" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_TradingSymbol_019e22b1-2149-7a3d-8d07-8196ee63b822" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_TradingSymbol"/>
    <link:presentationArc order="5" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_TradingSymbol_019e22b1-2149-7a3d-8d07-8196ee63b822" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_SecurityExchangeName_019e22b1-2149-74cd-98ed-47f659bb7d05" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_SecurityExchangeName"/>
    <link:presentationArc order="6" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_SecurityExchangeName_019e22b1-2149-74cd-98ed-47f659bb7d05" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityCentralIndexKey_019e22b1-2149-7671-ad34-1bbd67a8de4d" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityCentralIndexKey"/>
    <link:presentationArc order="7" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityCentralIndexKey_019e22b1-2149-7671-ad34-1bbd67a8de4d" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_DocumentType_019e22b1-2149-77b6-ae61-2575cb400278" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_DocumentType"/>
    <link:presentationArc order="8" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_DocumentType_019e22b1-2149-77b6-ae61-2575cb400278" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_DocumentPeriodEndDate_019e22b1-2149-7b93-a91b-4675eea45956" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_DocumentPeriodEndDate"/>
    <link:presentationArc order="9" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_DocumentPeriodEndDate_019e22b1-2149-7b93-a91b-4675eea45956" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityRegistrantName_019e22b1-2149-7aa7-bbac-e6498e46231c" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityRegistrantName"/>
    <link:presentationArc order="10" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityRegistrantName_019e22b1-2149-7aa7-bbac-e6498e46231c" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityIncorporationStateCountryCode_019e22b1-2149-76ab-abb4-d17868708add" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityIncorporationStateCountryCode"/>
    <link:presentationArc order="11" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityIncorporationStateCountryCode_019e22b1-2149-76ab-abb4-d17868708add" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityFileNumber_019e22b1-2149-7c2e-8a23-598f57c681cd" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityFileNumber"/>
    <link:presentationArc order="12" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityFileNumber_019e22b1-2149-7c2e-8a23-598f57c681cd" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityTaxIdentificationNumber_019e22b1-2149-73b9-b704-2693954bfa72" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityTaxIdentificationNumber"/>
    <link:presentationArc order="13" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityTaxIdentificationNumber_019e22b1-2149-73b9-b704-2693954bfa72" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressAddressLine1_019e22b1-2149-7605-a8a1-6c1600cccc22" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressAddressLine1"/>
    <link:presentationArc order="14" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityAddressAddressLine1_019e22b1-2149-7605-a8a1-6c1600cccc22" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressCityOrTown_019e22b1-2149-75de-90d6-223d819bc0cd" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressCityOrTown"/>
    <link:presentationArc order="15" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityAddressCityOrTown_019e22b1-2149-75de-90d6-223d819bc0cd" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressStateOrProvince_019e22b1-2149-7dbf-9f6a-a7903b10f3e1" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressStateOrProvince"/>
    <link:presentationArc order="16" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityAddressStateOrProvince_019e22b1-2149-7dbf-9f6a-a7903b10f3e1" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressPostalZipCode_019e22b1-2149-7d36-8f8f-5c4770e7168a" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressPostalZipCode"/>
    <link:presentationArc order="17" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_EntityAddressPostalZipCode_019e22b1-2149-7d36-8f8f-5c4770e7168a" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_LocalPhoneNumber_019e22b1-2149-7672-97fc-c5c70710599a" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_LocalPhoneNumber"/>
    <link:presentationArc order="18" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_LocalPhoneNumber_019e22b1-2149-7672-97fc-c5c70710599a" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_CityAreaCode_019e22b1-2149-7460-bc7e-8cba24b8793d" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_CityAreaCode"/>
    <link:presentationArc order="19" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_CityAreaCode_019e22b1-2149-7460-bc7e-8cba24b8793d" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_WrittenCommunications_019e22b1-2149-74db-8d4b-887d9f2d2a80" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_WrittenCommunications"/>
    <link:presentationArc order="20" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_WrittenCommunications_019e22b1-2149-74db-8d4b-887d9f2d2a80" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_SolicitingMaterial_019e22b1-2149-7a0c-995b-12b9fda088b2" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_SolicitingMaterial"/>
    <link:presentationArc order="21" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_SolicitingMaterial_019e22b1-2149-7a0c-995b-12b9fda088b2" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
    <link:loc xlink:type="locator" xlink:label="loc_dei_AmendmentFlag_019e22b1-2149-778f-b382-ab8621a22754" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_AmendmentFlag"/>
    <link:presentationArc order="22" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="loc_dei_DocumentInformationLineItems_019e22b1-2149-77ed-8147-bcd1419e2a4a" xlink:to="loc_dei_AmendmentFlag_019e22b1-2149-778f-b382-ab8621a22754" xlink:type="arc" preferredLabel="http://www.xbrl.org/2003/role/terseLabel"/>
  </link:presentationLink>
</link:linkbase>
