A search is presented for single production of a vector-like B quark decaying into a Standard Model b-quark and a Standard Model Higgs boson, which decays into a pair. The search is carried out in 139 fb⁻¹ of √s= 13 TeV proton-proton collision data collected by the ATLAS detector at the LHC between 2015 and 2018. No significant deviation from the Standard Model background prediction is observed, and mass-dependent exclusion limits at the 95% confidence level are set on the resonance production cross-section in several theoretical scenarios determined by the couplings c_W, c_Z and c_H between the B quark and the Standard Model W, Z and Higgs bosons, respectively. For a vector-like B occurring as an isospin singlet, the search excludes values of c_W greater than 0.45 for a B resonance mass (m_B) between 1.0 and 1.2 TeV. For 1.2 TeV < m_B < 2.0 TeV, c_W values larger than 0.50–0.65 are excluded. If the B occurs as part of a (B, Y) doublet, the smallest excluded c_Z coupling values range between 0.3 and 0.5 across the investigated resonance mass range 1.0 TeV < m_B < 2.0 TeV.