A coupling strategy between hybridizable discontinuous Galerkin (HDG) and continuous Galerkin (CG) methods is proposed in the framework of second-order elliptic operators. The coupled formulation is implemented and its convergence properties are established numerically by using manufactured solutions. Afterwards, the solution of the coupled Navier–Stokes convection–diffusion problem, using Boussinesq approximation, is formulated within the HDG framework and analysed using numerical experiments. Results of Rayleigh–Bénard convection flow are also presented and validated with literature data. Finally, the proposed coupled formulation between HDG and CG for heat equation is combined with the coupled Navier–Stokes/convection diffusion equations to formulate a new CG–HDG model for solving conjugate heat transfer problems. Benchmark examples are solved using the proposed model and validated with literature values. The proposed CG–HDG model is also compared with a CG–CG model, where all the equations are discretized using the CG method, and it is concluded that CG–HDG model can have a superior computational efficiency when compared to CG–CG model.