Federated learning (FL) enables privacy-preserving distributed machine learning by training models on edge client devices using their local data without revealing their raw data. In edge environments, various applications require different neural network models, making it crucial to perform joint training of multiple models on edge devices, known as multi-task FL. While existing multi-task FL approaches enhance resource utilization on edge devices through adaptive resource configuration or client selection, optimizing either of these aspects alone may lead to suboptimality. Therefore, in this paper, we explore a joint client selection and resource configuration method called JCSRC for multi-task FL, aiming to maximize energy efficiency in environments with limited computation and communication resources and heterogeneous client devices. Firstly, we formalize this problem as a mixed-integer nonlinear programming problem considering all these characteristics and prove its NP-hardness. To address this problem, we first design a multi-agent reinforcement learning (MARL)-based client selection method that selects appropriate clients for each task to train their models. The MARL method makes client selection decisions based on the clients’ data quality, energy efficiency, communication, and computation capacity to ensure fast convergence and energy efficiency. Then, we design a particle swarm optimization (PSO)-based resource configuration scheme that configures appropriate computation and bandwidth resources for each task on each client. The PSO scheme makes resource configuration decisions based on theoretically derived optimal CPU frequency and bandwidth to achieve high energy efficiency. Finally, we carry out extensive simulations and testbed-based experiments to validate our proposed JCSRC. The results demonstrate that, in comparison to state-of-the-art solutions, JCSRC can save energy consumption by up to 59% to achieve the target accuracy.