The variability of sea surface temperatures (SSTs) is crucial in climate dynamics, influencing marine ecosystems and human activities. This study leverages graph neural networks (GNNs), specifically a GraphSAGE model, to forecast SSTs and their anomalies (SSTAs), focusing on the global scale structure of climatological data. We introduce an improved graph construction technique for SST teleconnection representation. Our results highlight the GraphSAGE model's capability in 1‐month‐ahead global SST and SSTA forecasting, with SST predictions spanning up to 2 years with a recursive approach. Notably, regions with persistent currents exhibited enhanced SSTA predictability, contrasting with equatorial and Antarctic areas. Our GNN outperformed both the persistence model and traditional methods. Additionally, we offer an SST and SSTA graph data set based on ERA5 and a graph generation tool. This GNN case study has shown how the GraphSAGE can be used in SST and SSTA forecasting, and will provide a foundation for further refinements such as adjusting graph construction, optimizing imbalanced regression techniques for extreme SSTAs, and integrating GNNs with other temporal pattern learning methods to improve long‐term predictions.