The role of phasor measurement unit (PMU) data as real-time indicators of system dynamics is critically important for accurate state estimation in power systems. PMUs, being cyber-physical devices, are susceptible to cyberattacks, such as false data injection (FDI). As FDI can lead to incorrect state estimation and subsequent destructive impacts, the prompt detection of falsified data is crucial to preclude such adverse outcomes. In response to this challenge, this paper introduces a spatial-temporal graph neural network (ST-GNN) for the detection and localization of anomalies in the PMU network. The model incorporates a convolutional neural network and long short-term memory units, which are adept at extracting spatial and temporal features effectively. The inclusion of graph-based analysis in our model significantly improves the understanding of interconnections between neighboring PMUs, thereby enhancing its precision in detecting and pinpointing anomalies, even under sophisticated stealth false data injection attacks. The performance of this framework has been thoroughly evaluated on two IEEE test systems, the 39-bus and 127-bus systems, across a variety of attack scenarios. The results from these evaluations affirm the high accuracy of the model, highlighting its potential as a reliable tool for safeguarding power systems against cyber-physical threats.