We perform a theoretical and computational study of relativistic one-electron homonuclear diatomic quasimolecules subject to strong electromagnetic fields linearly polarized along the molecular axis. Several quasimolecules with the nuclear charges 1-92 and appropriately scaled internuclear distances and field parameters are used in the calculations. The time-dependent Dirac equation is solved with the help of the generalized pseudospectral method in prolate spheroidal coordinates. We have found that employing this coordinate system makes it possible to avoid emergence of spurious states, which usually show up when solving the Dirac equation numerically. For lower carrier frequencies, interaction with the driving field is described within the dipole approximation. Relativistic effects in the multiphoton ionization probabilities are investigated with respect to the internuclear distance in the quasimolecule. For higher frequencies, the interaction with the field is described beyond the dipole approximation. Nondipole effects in the ionization probability are discussed.