Stochastic games form the foundational mathematical framework for describing multiagent interactions and underpin the theoretical foundations of multiagent reinforcement learning (MARL) and optimal decision making. However, previous research has typically focused on either two-agent settings or large-scale well-mixed agent populations, where the considered interaction scenarios were far from realistic. In this article, we consider structured populations where agents can interact with immediate neighbors. By using the pair-approximation method, we develop a new dynamical model to describe the $Q$ -learning dynamics in stochastic games on regular graphs. Through comparisons with agent-based simulation results, we validate the accuracy of our dynamical model across various stochastic games, population structures, and algorithm parameters. Our research thus provides both qualitative and quantitative insights into the effects of state transition rules and graph topologies in population dynamics. In particular, we show that, under certain conditions, state transitions can significantly promote the evolution of cooperation in social dilemmas. We also explored the effects of agent degree on cooperation, and unlike previous findings, we show that this can have either positive or negative implications for cooperation depending on the transition rules.