Within a fully relativistic framework, we discuss the one‑loop self‑energy correction, generalized to the presence of external blackbody radiation, for a bound electron in the hydrogen molecular ion. The calculations rely on the thermal part of the photon propagator from finite-temperature quantum electrodynamics. The ADKB method, which extends the dual kinetic balance condition to axially symmetric systems, is adopted together with a B-spline-based trial function. The results are obtained within the Born–Oppenheimer adiabatic approximation. Thermal shift for the ground term of the hydrogen molecular ion is computed as a function of the internuclear distance. Including this correction, we present the resulting shifts of the vibrational levels. The analysis focuses on the dynamical effect, a comparison is made with data available in the current literature.