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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2017-10-19</periodOfReport>

    <notSubjectToSection16>0</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>94163</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>Preferred Shares</value>
            </securityTitle>
            <transactionDate>
                <value>2017-10-19</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>1000</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>100000.00</value>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>7730</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 1,000 preferred shares reported as disposed of in Table I represent variable rate demand preferred shares (the &quot;VRDP Shares&quot;) that were beneficially owned by Wells Fargo Municipal Capital Strategies, LLC (&quot;Capital Strategies&quot;). The VRDP Shares were disposed of as a result of a redemption by the Issuer for a redemption price of $100,000 per share. Capital Strategies is a wholly owned subsidiary of Wells Fargo &amp; Company (&quot;Wells Fargo&quot;).</footnote>
        <footnote id="F2">This statement is jointly filed by Wells Fargo, Capital Strategies and Wells Fargo Bank, National Association (&quot;WFBNA&quot;). 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 subsidiary Capital Strategies and WFBNA. 2,380 of the Securities are beneficially owned by Capital Strategies (comprised of 2,380 variable rate munifund term preferred shares of the Issuer), and 5,350 of the Securities are beneficially owned by WFBNA (comprised of 5,350 variable rate munifund term preferred shares of the Issuer). 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>/s/ Lori Ward</signatureName>
        <signatureDate>2018-03-13</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>/s/ Adam Joseph</signatureName>
        <signatureDate>2018-03-13</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>/s/ Adam Joseph</signatureName>
        <signatureDate>2018-03-13</signatureDate>
    </ownerSignature>
</ownershipDocument>
