Beam prediction and tracking (BPT) are key technology for millimeter wave communications. Typical techniques include Kalman filtering (KF) and Gaussian process (GP) regression. However, KF requires explicit system dynamics, which is difficult to obtain for complicated scenarios. In contrast, thanks to the data-driven manner, GP regression circumvents this challenging, which, however, suffers from prohibitive computational complexity. To tackle this issue, we propose a novel hybrid model and data driven approach, referred to as data-induced intelligent Kalman filtering (DIIKF). DIIKF learns the system dynamics via the data-driven manner, which can enjoy the advantages of both KF and GP while overcoming their drawbacks. In view that the system dynamics is available, we further propose long-term prediction and design an efficient algorithm. Simulation results show that our method approaches the optimal oracle solution (in terms of effective achievable rate), with the linear complexity order.