A search for flavor-changing neutral-current couplings between a top quark, an up or charm quark, and a $Z$ boson is presented, using proton--proton collision data at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ collected by the ATLAS detector at the Large Hadron Collider. The analyzed data set corresponds to an integrated luminosity of $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The search targets both single-top-quark events produced as $gq\ensuremath{\rightarrow}tZ$ (with $q=u$, $c$) and top-quark-pair events, with one top quark decaying through the $t\ensuremath{\rightarrow}Zq$ channel. The analysis considers events with three leptons (electrons or muons), a $b$-tagged jet, possible additional jets, and missing transverse momentum. The data are found to be consistent with the background-only hypothesis and 95% confidence-level limits on the $t\ensuremath{\rightarrow}Zq$ branching ratios, assuming only tensor operators of the Standard Model effective field theory framework contribute to the $tZq$ vertices. These are $6.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ ($13\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$) for $t\ensuremath{\rightarrow}Zu$ ($t\ensuremath{\rightarrow}Zc$) for a left-handed $tZq$ coupling, and $6.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ ($12\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$) in the case of a right-handed coupling. These results are interpreted as 95% CL upper limits on the strength of the corresponding couplings, yielding limits for $|{C}_{uW}^{(13)*}|$ and $|{C}_{uB}^{(13)*}|$ ($|{C}_{uW}^{(31)}|$ and $|{C}_{uB}^{(31)}|$) of 0.15 (0.16), and limits for $|{C}_{uW}^{(23)*}|$ and $|{C}_{uB}^{(23)*}|$ ($|{C}_{uW}^{(32)}|$ and $|{C}_{uB}^{(32)}|$) of 0.22 (0.21), assuming a new-physics energy scale ${\mathrm{\ensuremath{\Lambda}}}_{\mathrm{NP}}$ of 1 TeV.