The paper investigates stationary and nonstationary boundary value problems on a two-link thermal graph with local and coupled boundary conditions. A linear graph consisting of two sequentially connected rods with different thermal parameters and a single junction point is considered. Based on the generalized functions method and the Fourier transform with respect to time, a unified technique for solving boundary value problems on a two-link thermal graph is developed. Resolving systems of equations for the Dirichlet and Neumann–Dirichlet problems are constructed, and analytical integral representations of the solutions are obtained. Numerical experiments were carried out to study temperature distribution and the influence of oscillation frequency on the thermal state of the graph. The obtained results can be applied to modeling thermal processes in multilink rod structures used in construction, mechanical engineering, and thermal networks.