Federated learning (FL) can protect data privacy but has difficulties in motivating user equipment (UE) to engage in task training. This paper proposes a Bertrand-game based framework to address the incentive problem, where a model owner (MO) issues an FL task and the employed UEs help train the model by using their local data. Specially, we consider the impact of time-varying task load and channel quality on UE’s motivation to engage in the FL task. We adopt the finite-state discrete-time Markov chain (FSDT-MC) to predict these parameters during the FL task. Depending on the performance metrics set by the MO and the estimated energy cost of the FL task, each UE seeks to maximize its profit. We obtain the Nash equilibrium (NE) of the game in closed form, and develop a distributed iterative algorithm to find it. Finally, the simulation result verifies the effectiveness of the proposed approach.