In many human-computer interaction applications, fast and accurate hand tracking is necessary for an immersive experience. However, raw hand motion data can be flawed due to issues such as joint occlusions and high-frequency noise, hindering the interaction. Using only current motion for interaction can lead to lag, so predicting future movement is crucial for a faster response. Our solution is the Multi-task Spatial-Temporal Graph Auto-Encoder (Multi-STGAE), a model that accurately denoises and predicts hand motion by exploiting the inter-dependency of both tasks. The model ensures a stable and accurate prediction through denoising while maintaining motion dynamics to avoid over-smoothed motion and alleviate time delays through prediction. A gate mechanism is integrated to prevent negative transfer between tasks and further boost multi-task performance. Multi-STGAE also includes a spatial-temporal graph autoencoder block, which models hand structures and motion coherence through graph convolutional networks, reducing noise while preserving hand physiology. Additionally, we design a novel hand partition strategy and hand bone loss to improve natural hand motion generation. We validate the effectiveness of our proposed method by contributing two large-scale datasets with a data corruption algorithm based on two benchmark datasets. To evaluate the natural characteristics of the denoised and predicted hand motion, we propose two structural metrics. Experimental results show that our method outperforms the state-of-the-art, showcasing how the multi-task framework enables mutual benefits between denoising and prediction.