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