We study numerically the conjugate heat transfer between the walls of a parallel plate microchannel and a homogeneous porous medium fully saturated with a liquid that is found in motion due to an external pressure gradient. The origin of this problem is caused by a uniform heat flux imposed at the external surfaces of the walls of the microchannel that have a finite thermal conductivity. In this manner, the competition and heat transfer mechanisms between both regions characterized by the thermal resistances, conduct to a conjugate formulation that originates a dimensionless conjugate parameter ${\alpha }_c$. This parameter measures the ratio of both thermal resistances, and, for large values of this parameter, the longitudinal heat conduction effects in the walls are very important and suffer significant deviations when compared with the case with finite values for this parameter. The dimensionless governing equations for both regions are established with the corresponding boundary conditions, and the numerical results show that the aspect ratios of both regions, controlled through the dimensionless parameters ${ϵ}_h$ and $ϵ$, play an important role in distinguishing the presence of the longitudinal heat conduction effects in the walls. For instance, if the ratios ${\alpha }_c/{ϵ}_h^2=25$ and ${\alpha }_c/{ϵ}^2=1$, the longitudinal effects of heat transfer are very important in the walls of the microchannel, whereas in the porous matrix there are effects of heat transfer in both directions, whereas if ${\alpha }_c/{ϵ}_h^2={\alpha }_c/{ϵ}^2=25$ only the longitudinal conduction effects are significant for both regions.