Learning over massive data stored in different locations is essential in many real-world applications. However, sharing data is full of challenges due to the increasing demands of privacy and security with the growing use of smart mobile devices and IoT devices. Federated learning provides a potential solution to privacy-preserving and secure machine learning, by means of jointly training a global model without uploading data distributed on multiple devices to a central server. However, most existing work on federated learning adopts machine learning models with full-precision weights, and almost all these models contain a large number of redundant parameters that do not need to be transmitted to the server, consuming an excessive amount of communication costs. To address this issue, we propose a federated trained ternary quantization (FTTQ) algorithm, which optimizes the quantized networks on the clients through a self-learning quantization factor. Theoretical proofs of the convergence of quantization factors, unbiasedness of FTTQ, as well as a reduced weight divergence are given. On the basis of FTTQ, we propose a ternary federated averaging protocol (T-FedAvg) to reduce the upstream and downstream communication of federated learning systems. Empirical experiments are conducted to train widely used deep learning models on publicly available datasets, and our results demonstrate that the proposed T-FedAvg is effective in reducing communication costs and can even achieve slightly better performance on non-IID data in contrast to the canonical federated learning algorithms.