A search for pair-produced vector-like quarks using events with exactly one lepton (e or $\mu $), at least four jets including at least one b-tagged jet, and large missing transverse momentum is presented. Data from proton–proton collisions at a centre-of-mass energy of $\sqrt{s}=$13 $\text {TeV}$, recorded by the ATLAS detector at the LHC from 2015 to 2018 and corresponding to an integrated luminosity of 139 fb$^{-1}$, are analysed. Vector-like partners T and B of the top and bottom quarks are considered, as is a vector-like X with charge $+5/3$, assuming their decay into a W, Z, or Higgs boson and a third-generation quark. No significant deviations from the Standard Model expectation are observed. Upper limits on the production cross-section of T and B quark pairs as a function of their mass are derived for various decay branching ratio scenarios. The strongest lower limits on the masses are 1.59 $\text {TeV}$ assuming mass-degenerate vector-like quarks and branching ratios corresponding to the weak-isospin doublet model, and 1.47 $\text {TeV}$ (1.46 $\text {TeV}$) for exclusive $T \rightarrow Zt$ ($B/X \rightarrow Wt$) decays. In addition, lower limits on the T and B quark masses are derived for all possible branching ratios.