Numerical simulations of fluid-structure interaction problems is an arduous task, specifically for scenarios involving complex moving bodies. Immersed boundary (IB) method which has emerged in recent years as an efficient option for dealing with such hydrodynamic problems, offers a non-body conformal strategy for the smooth exchange of information across interface in moving boundary problems. Amongst several variants of IB with the lattice-Boltzmann (LB) method, the partially saturated cells (PSC) approach has proven to be a promising one, as it offers a unified evolution equation for all the media present in the computational domain. The PSC governing equation features a volume-of-body-based solid-collision operator, which accounts for the accurate implementation of the boundary conditions and dictates a smooth transition of flow variables between the solid and fluid cells. The application of PSC methodology, however, is mostly restricted to hydrodynamic problems till date. In the present study, we propose a novel solid collision operator to simulate incompressible thermal flows with moving bodies. We demonstrate that the PSC solver is as competitive as its existing counterpart and exhibits at least first-order accuracy, while also being Galilean-invariant and capable of accurately implementing the Dirichlet boundary condition. The present investigations further showcases the efficacy and robustness of the new collision operator and PSC approach using a wide spectrum of test cases, encompassing free, forced, and mixed convection problems, as well as both stationary and moving boundaries. The proposed algorithm therefore offers a promising and robust computational framework for complex fluid-particle interaction problems with heat transfer.