We investigate the process $e^{+}e^{-} rightarrow Kbar{K}J/psi$ at center-of-mass energies from 4.189 to 4.600 GeV using 4.7 fb$^{-1}$ of data collected by the BESIII detector at the BEPCII collider. The Born cross sections for the reactions $e^{+}e^{-} rightarrow K^{+}K^{-}J/psi$ and $K_{S}^{0} K_{S}^{0} J/psi$ are measured as a function of center-of-mass energy. The energy dependence of the cross section for $e^{+}e^{-} rightarrow K^{+}K^{-}J/psi$ is shown to differ from that for $pi^{+}pi^{-}J/psi$ in the region around the $Y(4260)$. In addition, there is evidence for a structure around 4.5 GeV in the $e^{+}e^{-} rightarrow K^{+}K^{-}J/psi$ cross section that is not present in $pi^{+}pi^{-}J/psi$.