A search is presented for single production of a vector-like quark (T) decaying to a Z boson and a top quark, with the Z boson decaying leptonically and the top quark decaying hadronically. The search uses data collected by the CMS experiment in proton-proton collisions at a center-of-mass energy of 13 TeV in 2016, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The presence of forward jets is a particular characteristic of single production of vector-like quarks that is used in the analysis. For the first time, different T quark width hypotheses are studied, from negligibly small to 30% of the new particle mass. At the 95% confidence level, the product of cross section and branching fraction is excluded above values in the range 0.26-0.04 pb for T quark masses in the range 0.7-1.7 TeV, assuming a negligible width. A similar sensitivity is observed for widths of up to 30% of the T quark mass. The production of a heavy Z' boson decaying to Tt, with T $\rightarrow$ tZ, is also searched for, and limits on the product of cross section and branching fractions for this process are set between 0.13 and 0.06 pb for Z' boson masses in the range from 1.5 to 2.5 TeV.