<?xml version="1.0" encoding="US-ASCII"?>
<!-- Generated by Broadridge Transform (tm) - http://www.broadridge.com -->
<!-- Created: Wed Oct 21 16:17:22 UTC 2026 -->
<xs:schema elementFormDefault="qualified" xmlns:srt-types="http://fasb.org/srt-types/2026" targetNamespace="http://fmbonline.com/20261023" xmlns:link="http://www.xbrl.org/2003/linkbase" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:us-types="http://fasb.org/us-types/2026" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xbrldt="http://xbrl.org/2005/xbrldt" xmlns:dtr="http://www.xbrl.org/dtr/type/2024-01-31" xmlns:xbrli="http://www.xbrl.org/2003/instance" xmlns:fmcb="http://fmbonline.com/20261023">
  <xs:annotation>
    <xs:appinfo>
      <link:linkbaseRef xlink:type="simple" xlink:href="fmcb-20261023_lab.xml" xlink:arcrole="http://www.w3.org/1999/xlink/properties/linkbase" xlink:role="http://www.xbrl.org/2003/role/labelLinkbaseRef"/>
      <link:linkbaseRef xlink:type="simple" xlink:href="fmcb-20261023_pre.xml" xlink:arcrole="http://www.w3.org/1999/xlink/properties/linkbase" xlink:role="http://www.xbrl.org/2003/role/presentationLinkbaseRef"/>
      <link:roleType roleURI="http://fmbonline.com/role/DocumentAndEntityInformation" id="DocumentAndEntityInformation">
        <link:definition>000 - Document - Document And Entity Information</link:definition>
        <link:usedOn>link:presentationLink</link:usedOn>
        <link:usedOn>link:definitionLink</link:usedOn>
        <link:usedOn>link:calculationLink</link:usedOn>
      </link:roleType>
    </xs:appinfo>
  </xs:annotation>
  <xs:import namespace="http://www.xbrl.org/2003/instance" schemaLocation="http://www.xbrl.org/2003/xbrl-instance-2003-12-31.xsd"/>
  <xs:import namespace="http://www.xbrl.org/2003/linkbase" schemaLocation="http://www.xbrl.org/2003/xbrl-linkbase-2003-12-31.xsd"/>
  <xs:import namespace="http://fasb.org/us-types/2026" schemaLocation="https://xbrl.fasb.org/us-gaap/2026/elts/us-types-2026.xsd"/>
  <xs:import namespace="http://www.xbrl.org/dtr/type/2024-01-31" schemaLocation="https://www.xbrl.org/dtr/type/2024-01-31/types.xsd"/>
  <xs:import namespace="http://xbrl.org/2005/xbrldt" schemaLocation="http://www.xbrl.org/2005/xbrldt-2005.xsd"/>
  <xs:import namespace="http://fasb.org/srt-types/2026" schemaLocation="https://xbrl.fasb.org/srt/2026/elts/srt-types-2026.xsd"/>
  <xs:import namespace="http://xbrl.sec.gov/sic/2026" schemaLocation="https://xbrl.sec.gov/sic/2026/sic-2026.xsd"/>
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  <xs:element name="DocumentAndEntityInformationAbstract" type="xbrli:stringItemType" abstract="true" xbrli:periodType="duration" nillable="true" substitutionGroup="xbrli:item" id="fmcb_DocumentAndEntityInformationAbstract"/>
</xs:schema>
