Resource allocation significantly impacts the performance of vehicle-to-everything (V2X) networks. Most existing algorithms for resource allocation are based on optimization or machine learning (e.g., reinforcement learning). In this paper, we explore resource allocation in a V2X network under the framework of federated reinforcement learning (FRL). On one hand, the usage of RL overcomes many challenges from the model-based optimization schemes. On the other hand, federated learning (FL) enables agents to deal with a number of practical issues, such as privacy, communication overhead, and exploration efficiency. The framework of FRL is then implemented by the inexact alternative direction method of multipliers (ADMM), where subproblems are solved approximately using policy gradients and accelerated by an adaptive step size calculated from their second moments. The developed algorithm, PASM, is proven to be convergent under mild conditions and has a nice numerical performance compared with some baseline methods for solving the resource allocation problem in a V2X network.