Currently, microfluidic technology is employed in miniaturized devices requiring thermal management optimization. Hence, thermal analysis must be conducted concurrently with comprehensive parametric studies, effectively managed through hybrid numerical-analytical methodologies. This article investigates conjugated internal convection in two-dimensional, steady-state parallel plate microchannels. Convection-diffusion equations are applied to both domains (solid wall and fluid), considering heat diffusion, slip flow and temperature jump conditions. The Generalized Integral Transform Technique (GITT) provides a systematic approach for solving boundary value heat transfer problems; indeed several previous studies employing GITT have focused on single-domain approaches for advection-diffusion problems. However, this article introduces a novel methodology for two-domain analysis. Additionally, the Finite Element Method (FEM) is employed for verification purposes to ensure the accuracy and robustness of the proposed approach. This investigation analyzed the convergence rate of the GITT and determined the temperature profile across two domains. A parametric analysis of the local Nusselt number was conducted for various wall-fluid configurations and Knudsen-Péclet number combinations. The findings were validated against FEM and previous studies and demonstrated the effectiveness of the multidomain formulation in addressing conjugate heat transfer in microchannels.