Collective excitations of many-body electron systems can carry internal structure, supporting novel quantum geometric and topological properties. Among these are a quantum geometric dipole (QGD), which for excitons have direct significance as an internal polarization. For interlayer excitons of a bilayer system, this represents an in-plane dipole moment, which can be used to drive them with in-plane electric fields. In this work, we consider counterflow electric currents associated with driven excitons in such a bilayer system as a probe of their QGD structure. As a simple but non-trivial example, we analyze a structure with a one-dimensional periodic potential in a strong perpendicular magnetic field. The resulting magnetoexciton bands host QGD structure that distinguishes it from the exciton QGD of a uniform system. To model exciton transport we adopt a Boltzmann approach that includes inter-band tunneling, allowing us to consider non-equilibrium momentum distributions that result from strong layer-antisymmetric driving fields. We show how linear response to a layer-symmetric component of the driving fields provide information about the QGD, and that the broad QGD structure of the exciton bands can be probed by the varying the layer-antisymmetric field. Our results demonstrate that counterflow conductivity serves as a tunable probe of the internal quantum geometric structure carried by the interlayer excitons, connecting transport to the quantum geometry of many-body excitations.