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Abstract: Junctions composed of excitonic condensate (EC) bilayers exhibit intriguing physics due to the presence of interlayer tunneling. Together with the excitonic superflow, this tunneling induces a special kind of topological object which is analogous to the Josephson vortices in extended superconducting Josephson junction. We name such objects the ``tunneling vortices.'' In this work we propose to characterize the topological properties of these vortices in the current-injecting scheme under which multiple vortices can easily be introduced through current injection from one end of the EC junctions. We find theoretically that current responses demonstrate the quantized nature of vortices: the ejected currents show periodicity as a function of injected currents(or vortex numbers). Moreover, the injected-ejected current difference shows plateaus corresponding to different vortex numbers when varying tunneling strength ${\mathrm{\ensuremath{\Delta}}}_{t}$ or superfluid density ${\ensuremath{\rho}}_{S}$. Since all parameters are experimentally accessible, we expect real-world demonstrations of these enthralling quantization behaviors to come in the near future.