  We study the relativistic dynamics of fermion-antifermion pairs in the
Bonnor-Melvin magnetic (BMM) spacetime in non-zero cosmology. We focus on the
$(1+2)$-dimensional cylindrically symmetric BMM-spacetime background. Within
the context of such a magnetized universe, we rigorously investigate the
fully-covariant two-body Dirac equation. We derive the corresponding radial
part of the equation and obtain an exact closed form analytical solution for
the problem at hand. Moreover, we report some intriguing parallels with
relativistic and non-relativistic quantum oscillators in flat spaces. Notably,
our findings suggest a compelling correlation between the cosmological constant
and the energy spectrum of fermion-antifermion systems, hinting at profound
connections between quantum realms and cosmology.
