A search for the flavour-changing neutral-current decay $t\to qZ$ is presented. Data collected by the ATLAS detector during 2012 from proton-proton collisions at the Large Hadron Collider at a centre-of-mass energy of $\sqrt{s}=8$ TeV, corresponding to an integrated luminosity of 20.3 fb$^{-1}$, are analysed. Top-quark pair-production events with one top quark decaying through the $t\to qZ$ ($q=u,c$) channel and the other through the dominant Standard Model mode $t\to bW$ are considered as signal. Only the decays of the $Z$ boson to charged leptons and leptonic $W$ boson decays are used. No evidence for a signal is found and an observed (expected) upper limit on the $t\to qZ$ branching ratio of $7\times 10^{-4}$ ($8\times 10^{-4}$) is set at the 95% confidence level.