<p>This paper establishes a dynamic quantum Cournot game model incorporating an externality cost function, where participants aim to maximize relative profits and possess bounded rationality, updating their output for the next period using a gradient adjustment mechanism. Based on the established model, we analyze the existence and stability of quantum Nash equilibria and investigate the complex behavior of the system. The research results indicate that as the adjustment speed increases, the system’s stability decreases due to the emergence of Flip and Neimark–Sacker bifurcations. However, increasing the degree of quantum entanglement can delay the occurrence of bifurcation behavior. Furthermore, we find that when enterprises cannot control chaotic states by adjusting external cost parameters, they can transition the system from a chaotic to a stable state by altering product differentiation and quantum entanglement. Finally, numerical simulations validate the theoretical analysis and visually demonstrate complex dynamic characteristics, such as bifurcation diagrams, the maximum Lyapunov exponent, strange attractors, sensitivity to initial conditions, and chaos.</p>