This Letter presents the first study of Higgs boson production in association with a vector boson (V=W or Z) in the fully hadronic qqbb final state using data recorded by the ATLAS detector at the LHC in proton-proton collisions at sqrt[s]=13  TeV and corresponding to an integrated luminosity of 137  fb^{-1}. The vector bosons and Higgs bosons are each reconstructed as large-radius jets and tagged using jet substructure techniques. Dedicated tagging algorithms exploiting b-tagging properties are used to identify jets consistent with Higgs bosons decaying into bb[over ¯]. Dominant backgrounds from multijet production are determined directly from the data, and a likelihood fit to the jet mass distribution of Higgs boson candidates is used to extract the number of signal events. The VH production cross section is measured inclusively and differentially in several ranges of Higgs boson transverse momentum: 250-450, 450-650, and greater than 650 GeV. The inclusive signal yield relative to the standard model expectation is observed to be μ=1.4_{-0.9}^{+1.0} and the corresponding cross section is 3.1±1.3(stat)_{-1.4}^{+1.8}(syst)  pb.