A high-order unified solution method is presented for simulating the coupled fluid flow and heat transfer phenomena in solid materials. The method integrates the energy transfer processes of fluid and solid by transforming the compressible Navier-Stokes equations into a dimensionless system through a double-time-scale approach. A time scaling factor is introduced into the system to bridge and expedite the energy transfer processes between the mediums, thereby accelerating convergence towards a global steady state. To ensure consistent accuracy across material interfaces, high-order formulations are introduced to obtain the interface temperature and heat flux. The effectiveness of the methods are verified and validated through numerical experiments with a three-dimensional discontinuous Galerkin flow solver. Numerical results demonstrate the globally high-order accuracy of the unified method for fluid-solid conjugate heat transfer problems, enabling rapid and robust convergence with large CFL numbers of up to $10^8.$