The OH+ ion is an important constituent of the interstellar medium (ISM). It can be used as a probe of cosmic ray and X-ray ionization rates in molecular clouds as well as a tracer of oxygen chemistry. The deuterated variant of OH+, the OD+ ion, may also be present in the ISM despite the fact that it has not been detected yet. In this paper, we aim at providing quantitative insight into the OD+ chemistry and at accurately studying the possible formation and destruction processes of OD+ in the ISM. We study the formation and destruction of OD+ through the O+ + HD → OD+ + H and OH+ + D ↔ OD+ + H reactions that can occur in diffuse ISM. Reactive rate constants have been obtained from exact state-to-state quantum wave packet calculations for temperatures ranging from 10 to 1000 K. The new theoretical data are validated through a detailed comparison with available experimental data. The formation of OD+ is found to be less efficient than that of OH+. As a first application, the OD+/OH+ abundance ratio in ISM has been evaluated from a simple astrochemical model, and we found that this ratio can be larger than D/H abundance ratio only at low temperatures. These calculations may help in an astrochemical search of OD+ in cold ISM.