A search is performed for electroweak production of a vector-like top quark partner T of charge 2/3 in association with a top or bottom quark, using proton-proton collision data at $\sqrt{s} =$ 13 TeV collected by the CMS experiment at the LHC in 2016. The data sample corresponds to an integrated luminosity of 35.9 fb$^{-1}$. The search targets T quarks over a wide range of masses and fractional widths, decaying to a top quark and either a Higgs boson or a Z boson in fully hadronic final states. The search is performed using two experimentally distinct signatures that depend on whether or not each quark from the decays of the top quark, Higgs boson, or Z boson produces an individual resolved jet. Jet substructure, b tagging, and kinematic variables are used to identify the top quark and boson jets, and also to suppress the standard model backgrounds. The data are found to be consistent with the expected backgrounds. Upper limits at 95% confidence level are set on the cross sections for T quark-mediated production of tHQq, tZQq, and their sum, where Q is the associated top or bottom heavy quark and q is another associated quark. The limits are given for each search signature for various T quark widths up to 30% of the T quark mass, and are between 2 pb and 20 fb for T quark masses in the range 0.6-2.6 TeV. These results are significantly more sensitive than prior searches for electroweak single production of T$\to$tH and represent the first constraints on T$\to$tZ using hadronic decays of the Z boson with this production mode.