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 $\sqrt{s}=13\mathrm{TeV}$ collected by the ATLAS detector at the Large Hadron Collider. The analyzed data set corresponds to an integrated luminosity of $139{\mathrm{fb}}^{-1}$. The search targets both single-top-quark events produced as $gq\to tZ$ (with $q=u$, $c$) and top-quark-pair events, with one top quark decaying through the $t\to Zq$ 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 $t\to Zq$ branching ratios, assuming only tensor operators of the Standard Model effective field theory framework contribute to the $tZq$ vertices. These are $6.2\times {10}^{-5}$ ($13\times {10}^{-5}$) for $t\to Zu$ ($t\to Zc$) for a left-handed $tZq$ coupling, and $6.6\times {10}^{-5}$ ($12\times {10}^{-5}$) 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 $\left|C_{uW}^{\left(13\right)*}\right|$ and $\left|C_{uB}^{\left(13\right)*}\right|$ ($\left|C_{uW}^{\left(31\right)}\right|$ and $\left|C_{uB}^{\left(31\right)}\right|$) of 0.15 (0.16), and limits for $\left|C_{uW}^{\left(23\right)*}\right|$ and $\left|C_{uB}^{\left(23\right)*}\right|$ ($\left|C_{uW}^{\left(32\right)}\right|$ and $\left|C_{uB}^{\left(32\right)}\right|$) of 0.22 (0.21), assuming a new-physics energy scale ${\Lambda }_{\mathrm{NP}}$ of 1 TeV.