This paper studies the numerical simulation of the solution to the McKean-Vlasov equation with common noise. We begin by discretizing the solution in time using the Euler scheme, followed by spatial discretization through the particle method, inspired by the propagation of chaos property. Assuming Hölder continuity in time, as well as Lipschitz continuity in the state and measure arguments of the coefficient functions, we establish the convergence rate of the Euler scheme and the particle method. These results extend those for the standard McKean-Vlasov equation without common noise. Finally, we present two simulation examples : a modified conditional Ornstein Uhlenbeck process with common noise and an interbank market model.