Flavor-changing neutral currents are not present in the Standard Model at tree level and are suppressed in loop processes by the unitarity of the Cabibbo-Kobayashi-Maskawa matrix; the corresponding rates for top quark decay processes are experimentally unobservable. Extensions of the Standard Model can generate new flavor-changing neutral current processes, leading to signals which, if observed, would be unambiguous evidence of new interactions. A data set corresponding to an integrated luminosity of $36.1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $pp$ collisions at a center-of-mass energy of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ recorded with the ATLAS detector at the Large Hadron Collider is used to search for top quarks decaying to up or charm quarks with the emission of a Higgs boson, with subsequent Higgs boson decay to final states with at least one electron or muon. No signal is observed and limits on the branching fractions $\mathcal{B}(t\ensuremath{\rightarrow}Hc)<0.16%$ and $\mathcal{B}(t\ensuremath{\rightarrow}Hu)<0.19%$ at 95% confidence level are obtained (with expected limits of 0.15% in both cases).