A search for pair-produced heavy vectorlike charge-$2/3$ quarks, $T$, in $pp$ collisions at a center-of-mass energy of 7 TeV, is performed with the CMS detector at the LHC. Events consistent with the flavor-changing-neutral-current decay of a $T$ quark to a top quark and a $Z$ boson are selected by requiring two leptons from the $Z$-boson decay, as well as an additional isolated charged lepton. In a data sample corresponding to an integrated luminosity of $1.14\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$, the number of observed events is found to be consistent with the standard model background prediction. Assuming a branching fraction of 100% for the decay $T\ensuremath{\rightarrow}tZ$, a $T$ quark with a mass less than $475\text{ }\text{ }\mathrm{GeV}/{c}^{2}$ is excluded at the 95% confidence level.