In this work we propose and analyze a discontinuous Galerkin method for the nonlinear coupled Navier-Stokes/temperature (or Boussinesq) equations with Navier-type boundary condition for the velocity. Existence and uniqueness of the solution are obtained under a small data condition. We provide a priori error estimates in terms of a natural energy norms for the velocity, the pressure and the temperature. To our knowledge, it is the first time that a discontinuous Galerkin approximation for the full nonlinear coupled Bousinessq system, with Navier-type boundary condition for the velocity, is proposed and completely analyzed in both, continuous and discrete settings.