Currently, most medical institutions face the challenge of training a unified model using fragmented and isolated data to address disease prediction problems. Although federated learning has become the recognized paradigm for privacy-preserving model training, how to integrate federated learning with fMRI temporal characteristics to enhance predictive performance remains an open question for functional disease prediction. To address this challenging task, we propose a novel Federated Graph Spatio-Temporal (FedGST) framework for brain functional disease prediction. Specifically, anchor sampling is used to process variable-length time series data on local clients. Then dynamic functional connectivity graphs are generated via sliding windows and Pearson correlation coefficients. Next, we propose an InceptionTime model to extract temporal information from the dynamic functional connectivity graphs on the local clients. Finally, the hidden activation variables are sent to a global server. We propose a UniteGCN model on the global server to receive and process the hidden activation variables from clients. Then, the global server returns gradient information to clients for backpropagation and model parameter updating. Client models aggregate model parameters on the local server and distribute them to clients for the next round of training. We demonstrate that FedGST outperforms other federated learning methods and baselines on ABIDE-1 and ADHD200 datasets.