A secure vehicular communication system is crucial for preventing vehicle collisions on foggy highways; however, unconditionally powerful attackers can be the reason for vehicle chaos and accidents. Quantum cryptographic protocols offer a solution by leveraging the principles of quantum mechanics to provide inherently secure vehicular communication. The existing protocols cannot prevent quantum computer attacks. These protocols also require more computation costs. In this paper, we proposed a quantum protocol for vehicular communication based on Nash equilibrium, which can prevent quantum computer attacks. We use the Nash equilibrium to resist the collusion attack. In this protocol, we considered quantum attacks and computation costs as the performance parameters. The computation cost of this protocol is $O(1)$ as the TCS only needs to perform a one-time computation for sharing the secret signal. In our protocol, the share of secret signals is regularly transmitted to the vehicles, and vehicles jointly compute the secret signals to avoid vehicle chaos and accidents. The proposed protocol can be deployed on foggy highways to avoid vehicle chaos and accidents.