In this paper, we present an SQP-type proximal gradient method (SQP-PG) for composite optimization problems with equality constraints. At each iteration, SQP-PG solves a subproblem to get the search direction, and takes an exact penalty function as the merit function to determine if the trial step is accepted. The global convergence of the SQP-PG method is proved and the iteration complexity for obtaining an $\epsilon$-stationary point is analyzed. We also establish the local linear convergence result of the SQP-PG method under the second-order sufficient condition. Numerical results demonstrate that, compared to the state-of-the-art algorithms, SQP-PG is an effective method for equality constrained composite optimization problems.