We study numerical methods for the time-dependent magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation, in which the Joule effect and Viscous heating are taken into account. To overcome the difficulties of very low regularity of the heat source terms, a regularized weak system is proposed to deal with Joule and Viscous heating terms. We consider an Euler semi-implicit semi-discrete scheme for the regularized system. As both discrete parameter and regularization parameter tend to zero, we prove that the discrete solution converges to a weak solution of the original problem. Next, we consider the fully discrete Euler semi-implicit scheme based on the mixed finite method to approximate the fluid equation and Nedelec edge element to the magnetic induction. The fully discrete scheme requires only solving a linear system per time step. The error estimates for the velocity, magnetic induction and temperature are derived under a proper regularity assumption for the exact solution. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.