We prove that an implicit time Euler scheme for the 2D-Boussinesq model on the torus $D$ converges. Various moment of the $W^{1,2}$-norms of the velocity and temperature, as well as their discretizations, are computed. We obtain the optimal speed of convergence in probability, and a logarithmic speed of convergence in $L^2(\Omega)$. These results are deduced from a time regularity of the solution both in $L^2(D)$ and $W^{1,2}(D)$, and from an $L^2(\Omega)$ convergence restricted to a subset where the $W^{1,2}$-noms of the solutions are bounded.