This paper reports constraints on Higgs boson production with transverse momentum above 1 TeV. The analyzed data from proton-proton collisions at a center-of-mass energy of 13 TeV were recorded with the ATLAS detector at the Large Hadron Collider from 2015 to 2018 and correspond to an integrated luminosity of 136 fb(-1.) Higgs bosons decaying into b (b) over bar are reconstructed as single large-radius jets recoiling against a hadronic system and are identified by the experimental signature of two b-hadron decays. The experimental techniques are validated in the same kinematic regime using the Z -> b (b) over bar process. The 95% confidence-level upper limit on the cross section for Higgs boson production with transverse momentum above 450 GeV is 115 fb, and above 1 TeV it is 9.6 fb. The Standard Model cross section predictions for a Higgs boson with a mass of 125 GeV in the same kinematic regions are 18.4 fb and 0.13 fb, respectively.