Three of the most significant measured deviations from standard model predictions, the enhanced decay rate for $B\ensuremath{\rightarrow}{D}^{(*)}\ensuremath{\tau}\ensuremath{\nu}$, hints of lepton universality violation in $B\ensuremath{\rightarrow}{K}^{(*)}\ensuremath{\ell}\ensuremath{\ell}$ decays, and the anomalous magnetic moment of the muon, can be explained by the existence of leptoquarks (LQs) with large couplings to third-generation quarks and masses at the TeV scale. The existence of these states can be probed at the LHC in high energy proton-proton collisions. A novel search is presented for pair production of LQs coupled to a top quark and a muon using data at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of $35.9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$, recorded by the CMS experiment. No deviation from the standard model prediction has been observed and scalar LQs decaying exclusively into $t\ensuremath{\mu}$ are excluded up to masses of 1420 GeV. The results of this search are combined with those from previous searches for LQ decays into $t\ensuremath{\tau}$ and $b\ensuremath{\nu}$, which excluded scalar LQs below masses of 900 and 1080 GeV. Vector LQs are excluded up to masses of 1190 GeV for all possible combinations of branching fractions to $t\ensuremath{\mu}$, $t\ensuremath{\tau}$ and $b\ensuremath{\nu}$. With this analysis, all relevant couplings of LQs with an electric charge of $\ensuremath{-}1/3$ to third-generation quarks are probed for the first time.