We explore the use of policy approximations to reduce the computational cost of learning Nash equilibria in zero-sum stochastic games. We propose a new Q-learning type algorithm that uses a sequence of entropy-regularized soft policies to approximate the Nash policy during the Q-function updates. We prove that under certain conditions, by updating the entropy regularization, the algorithm converges to a Nash equilibrium. We also demonstrate the proposed algorithm's ability to transfer previous training experiences, enabling the agents to adapt quickly to new environments. We provide a dynamic hyper-parameter scheduling scheme to further expedite convergence. Empirical results applied to a number of stochastic games verify that the proposed algorithm converges to the Nash equilibrium, while exhibiting a major speed-up over existing algorithms.