Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams. The main goal of this talk is to describe how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to the Ehrhart (quasi-)polynomial of its toric diagram. This is part of joint work with Leonardo Macarini and Miguel Moreira (arXiv:2202.00442).