A search for leptoquarks produced singly and in pairs in proton-proton collisions is presented. We consider the leptoquark (LQ) to be a scalar particle of charge -1/3$e$ coupling to a top quark plus a tau lepton ($\mathrm{t}\tau$) or a bottom quark plus a neutrino ($\mathrm{b}\nu$), or a vector particle of charge +2/3$e$, coupling to $\mathrm{t}\nu$ or $\mathrm{b}\tau$. These choices are motivated by models that can explain a series of anomalies observed in the measurement of B meson decays. In this analysis the signatures $\mathrm{t}\tau\nu\mathrm{b}$ and $\mathrm{t}\tau\nu$ are probed, using data recorded by the CMS experiment at the CERN LHC at $\sqrt{s} =$ 13 TeV and that correspond to an integrated luminosity of 137 fb$^{-1}$. These signatures have not been previously explored in a dedicated search. The data are found to be in agreement with the standard model prediction. Lower limits at 95% confidence level are set on the LQ mass in the range 0.98-1.73 TeV, depending on the LQ spin and its coupling $\lambda$ to a lepton and a quark, and assuming equal branching fractions for the two LQ decay modes considered. These are the most stringent constraints to date on the existence of leptoquarks in this scenario.