Federated learning is a distributed machine learning method that allows numerous clients to cooperatively train AI models without uploading their raw data. However, the heterogeneous data will significantly affect the performance of the global model, and self-interested clients are often reluctant to participate in learning tasks without satisfactory rewards. To tackle these challenges, this paper proposes a novel incentive mechanism, called Qualiaty-Aware Reverse Auction (QARA), to ensure that clients can maximize their own utilities when conducting honest bidding. Specifically, we first use Shannon entropy to combine data quantity and diversity to measure the quality of client data, allowing clients to be selected and incentivized based on their data quality and bids within a limited budget. Next, we formalize the data quality maximization problem with the aim of maximizing the quality of the selected client data, and show that it is NP-hard. Furthermore, to solve such a intractable problem, we design a greedy algorithm with an approximation ratio of 1/2. Theoretically, we prove that QARA satisfies dominant strategy incentive compatibility, computational efficiency, individual rationality, and budget feasibility. Last, we evaluate the generalization of QARA and four baselines on five real-world datasets to demonstrate the superiority of QARA.