The relativistic quantum motion of scalar bosons under the influence of a full vector (minimal Aμ and nonminimal Xμ) and scalar (Vs) interactions embedded in the background of a cosmic string is explored in the context of the Klein–Gordon equation. Considering Coulomb interactions, the effects of this topological defect in equation of motion, phase shift and S-matrix are analyzed and discussed. Bound-state solutions are obtained from poles of the S-matrix and it is shown that bound-state solutions are possible only for a restrict range of coupling constants.