Quantum Federated Learning (QFL) has emerged as a promising research direction by combining the strengths of quantum computing and federated learning. However, existing QFL solutions have consistently failed to simultaneously improve client training efficiency and ensure communication security. In this paper, we present a novel Multi-qubit Broadcast-based QFL framework (MB-QFL) to address the efficiency and security challenges of existing approaches. The framework employs a novel multi-qubit broadcast protocol and a quantum average method to secure the information transmission process. The multi-qubit broadcast protocol overcomes the limitations of existing protocols by allowing the transmission of an arbitraryS-qubit state from one sender to multiple (Q) receivers, whereas earlier protocols were restricted to broadcast one or two qubit state to recipients. Additionally, we propose an averaging method for quantum states, which exploits the probabilistic cloning technique to achieve aggregation in MB-QFL. The security analysis demonstrates that MB-QFL can effectively protect against inference attacks from malicious clients, as well as eavesdropping and intercept-and-resend attacks during communication. The algorithm complexity of MB-QFL is significantly lower than existing QFLs. Besides, the experimental results indicate that MB-QFL achieves higher classification accuracy than other QFLs.