The associated production of a Higgs boson with a W or Z boson decaying into leptons and where the Higgs boson decays to a b (b) over bar pair is measured in the high vector-boson transverse momentum regime, above 250 GeV, with the ATLAS detector. The analysed data, corresponding to an integrated luminosity of 139 fb(-1), were collected in proton-proton collisions at the Large Hadron Collider between 2015 and 2018 at a centre-of-mass energy of root s = 13 TeV. The measured signal strength, defined as the ratio of the measured signal yield to that predicted by the Standard Model, is 0.72(-0.36)(+0.39) corresponding to an observed (expected) significance of 2.1 (2.7) standard deviations. Cross-sections of associated production of a Higgs boson decaying into b quark pairs with a W or Z gauge boson, decaying into leptons, are measured in two exclusive vector boson transverse momentum regions, 250-400 GeV and above 400 GeV, and interpreted as constraints on anomalous couplings in the framework of a Standard Model effective field theory.