<p>The development of quantum information theory based on quantum entangled states is advancing rapidly and has promoted the emergence of quantum game theory. Quantum game theory is quantum information processing applied to classical games, which solves problems that classical games cannot solve. This paper explores the quantum Cournot-Bertrand game model of two firms in a duopoly market with bounded rationality and product substitutability. The first and second firms choose production output and sales price as their decision variables, respectively. We first define the quantum Cournot-Bertrand dynamical system and study the existence and locally stable conditions of equilibrium points. Then we explore bifurcation conditions and bifurcation types. Finally, we do the bifurcation analysis and explore the impacts of the model parameters as the potential market size of products, the two firms' linear marginal cost, the players' speed of strategy adjustment, the product substitution index, and the quantum entanglement degree on the Nash equilibrium stability and bifurcation using numerical analysis and simulation, and provide economic explanations for the results.</p>