A new implementation of the finite amplitude method (FAM) for the solution of the relativistic quasiparticle random-phase approximation (RQRPA) is presented, based on the relativistic Hartree-Bogoliubov (RHB) model for deformed nuclei. The numerical accuracy and stability of the FAM–RQRPA is tested in a calculation of the monopole response of ²²O. As an illustrative example, the model is applied to a study of the evolution of monopole strength in the chain of Sm isotopes, including the splitting of the giant monopole resonance in axially deformed systems.