In this paper, we discuss the problem of a class of linear quadratic zero-sum stochastic differential games with Markov regime switching in continuous time. Under the  condition of generalized It\^{o}'s differential rule, by  introducing a generalized Riccati differential (algebraic) equation,  it is proved that the solvability of the associated generalized Riccati  equation is both sufficient and necessary condition for the existence   of equilibrium strategies, meanwhile, the explicit solution of equilibrium strategies with closed form and the optimal value of cost functional are obtained. Fi ally, a numerical example is given to illustrate the validity of the obtained results.