Centralized machine learning algorithms in vehicular networks face privacy and resource constraints. Federated Learning (FL) addresses these by enabling collaborative model training without sharing raw data. To incentivize vehicle participation despite resource constraints, we propose a contract between Roadside Units (RSUs) and vehicles, specifying required computation resources and rewards based on local data quality. The contract is designed to maximize the utility of the RSU, subject to constraints that ensure the participation of the vehicles in the task and truthful reporting of their data quality parameter. However, designing an optimal contract with classical methods is limited to specific utility functions. To extend the design methodology of optimal contracts to a broader range of utility functions, our study proposes a deep learning-based solution. Specifically, we present a novel application of Deep Neural Networks (DNNs), marking the first instance of applying DNNs in contract theory-based incentive mechanisms to achieve nearoptimal contracts. We design the DNN's structure to inherently satisfy specific constraints of the optimization problem, enhancing its efficiency and accuracy in determining the optimal contract. We utilize an augmented Lagrangian method to approximate the optimal contract in federated learning tasks. A case study on traffic sign recognition demonstrates the applicability and effectiveness of our method. Comparing our approach with the well-known generative diffusion model shows that in the FL scenarios, the DNN results are closer to the optimal contract compared to those of the generative diffusion model.