A search for the production of a single top quark in association with a Z boson is presented, both to identify the expected standard model process and to search for flavour-changing neutral current interactions. The data sample corresponds to an integrated luminosity of 19.7 fb$^{−1}$ recorded by the CMS experiment at the LHC in proton-proton collisions at $ \sqrt{s}=8 $ TeV. Final states with three leptons (electrons or muons) and at least one jet are investigated. An events yield compatible with tZq standard model production is observed, and the corresponding cross section is measured to be σ(pp → tZq → ℓνbℓ$^{+}$ ℓ$^{−}$q) = 10$_{− 7}^{+ 8}$ fb with a significance of 2.4 standard deviations. No presence of flavour-changing neutral current production of tZq is observed. Exclusion limits at 95% confidence level on the branching fractions of a top quark decaying to a Z boson and an up or a charm quark are found to be ℬ(t → Zu) < 0.022% and ℬ(t → Zc) < 0.049%.