<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2020-09-30</periodOfReport>

    <notSubjectToSection16>1</notSubjectToSection16>

    <issuer>
        <issuerCik>0001195737</issuerCik>
        <issuerName>Nuveen AMT-Free Quality Municipal Income Fund</issuerName>
        <issuerTradingSymbol>NEA</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0000072971</rptOwnerCik>
            <rptOwnerName>WELLS FARGO &amp; COMPANY/MN</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>420 MONTGOMERY STREET</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>SAN FRANCISCO</rptOwnerCity>
            <rptOwnerState>CA</rptOwnerState>
            <rptOwnerZipCode>94104</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001585457</rptOwnerCik>
            <rptOwnerName>Wells Fargo Municipal Capital Strategies, LLC</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>375 PARK AVENUE</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10152</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0000740906</rptOwnerCik>
            <rptOwnerName>WELLS FARGO BANK N A</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>101 NORTH PHILLIPS AVENUE</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>SIOUX FALLS</rptOwnerCity>
            <rptOwnerState>SD</rptOwnerState>
            <rptOwnerZipCode>57104</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>MuniFund Preferred Shares</value>
            </securityTitle>
            <transactionDate>
                <value>2020-09-30</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>J</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
                <footnoteId id="F2"/>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>2380</value>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>3350</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>By Subsidiary</value>
                    <footnoteId id="F2"/>
                    <footnoteId id="F3"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">The 2,380 preferred shares reported as disposed of in Table I represent MuniFund Preferred Shares, Series C (&quot;MFP Shares&quot;) that were beneficially owned by Wells Fargo Municipal Capital Strategies, LLC (&quot;Capital Strategies&quot;). The MFP Shares were disposed of as a result of a sale of the MFP Shares in an open market transaction for a price of $238,000,000. Wells Fargo Bank, National Association (&quot;WFBNA&quot;) continues to own 3,350 MuniFund Preferred Shares, Series B of the Issuer.</footnote>
        <footnote id="F2">This statement is jointly filed by Wells Fargo &amp; Company (&quot;Wells Fargo&quot;), Capital Strategies and WFBNA. Wells Fargo holds an indirect interest in the securities listed in Table I (the &quot;Securities&quot;) by virtue of its indirect ownership of its subsidiaries Capital Strategies and WFBNA. Capital Strategies and WFBNA are each indirect wholly owned subsidiaries of Wells Fargo.</footnote>
        <footnote id="F3">Each reporting person declares that neither the filing of this statement nor anything herein shall be construed as an admission that such person is, for the purposes of Section 13(d) of the US Securities Exchange Act of 1934 or any other purpose, (i) acting (or has agreed or is agreeing to act together with any other person) as a partnership, limited partnership, syndicate or other group for the purpose of acquiring, holding or disposing of securities of the Issuer or otherwise with respect to the Issuer or any securities of the Issuer or (ii) a member of any group with respect to the Issuer or any securities of the Issuer.</footnote>
    </footnotes>

    <remarks>Exhibits Index
Exhibit 99.1 - Joint Filing Agreement
Exhibit 99.2 - Joint Filer Information</remarks>

    <ownerSignature>
        <signatureName>WELLS FARGO &amp; COMPANY, by: /s/ Patricia Arce</signatureName>
        <signatureDate>2020-10-02</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>WELLS FARGO MUNICIPAL CAPITAL STRATEGIES, LLC, by: /s/ Alejandro Piekarewicz</signatureName>
        <signatureDate>2020-10-02</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>WELLS FARGO BANK, NATIONAL ASSOCIATION, by: /s/ Alejandro Piekarewicz</signatureName>
        <signatureDate>2020-10-02</signatureDate>
    </ownerSignature>
</ownershipDocument>
