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 $ b\overline{b} $ pair. The search is carried out in 139 fb$^{−1}$ of $ \sqrt{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.