We investigate the dynamics of a composite system (CS) consisting of an interacting fermion-antifermion pair in the three-
dimensional spacetime background generated by a static point source. By considering the interaction between the particles as
Dirac oscillator coupling, we analyze the effects of spacetime topology on the energy of such a CS. To achieve this, we solve the
corresponding form of a two-body Dirac equation (fully-covariant) by assuming the center of mass of the particles is rest and locates
at the origin of the spatial geometry. Under this assumption, we arrive at a non-perturbative energy spectrum for the system in
question. This spectrum includes spin coupling and depends on the angular deficit parameter (ADP) of the geometric background.
This provides a suitable basis to determine the effects of the geometic background on the energy of the CS under consideration.
Our results show that such a CS behaves like a single quantum oscillator. Then, we analyze the alterations in the energy levels and
discuss the limits of the obtained results. We show that the effects of the geometric background on each energy level are not same
and there can be degeneracy in the energy levels for small values of the ADP.