We study Concave Constrained Markov Decision Processes (Concave CMDPs) where both the objective and constraints are defined as concave functions of the state-action occupancy measure. We propose the Variance-Reduced Primal-Dual Policy Gradient Algorithm (VR-PDPG), which updates the primal variable via policy gradient ascent and the dual variable via projected sub-gradient descent. Despite the challenges posed by the loss of additivity structure and the nonconcave nature of the problem, we establish the global convergence of VR-PDPG by exploiting a form of hidden concavity. In the exact setting, we prove an O(T-1/3) convergence rate for both the average optimality gap and constraint violation, which further improves to O(T-1/2) under strong concavity of the objective in the occupancy measure. In the sample-based setting, we demonstrate that VR-PDPG achieves an O(ε-4) sample complexity for ε-global optimality. Moreover, by incorporating a diminishing pessimistic term into the constraint, we show that VR-PDPG can attain a zero constraint violation without compromising the convergence rate of the optimality gap. Finally, we validate our methods through numerical experiments.