Geomagnetic storms, caused by solar wind energy transfer to Earth's magnetic field, can disrupt critical infrastructure like GPS, satellite communications, and power grids. The disturbance storm-time (Dst) index measures storm intensity. Despite advancements in empirical, physics-based, and machine-learning models using real-time solar wind data, accurately forecasting extreme geomagnetic events remains challenging due to noise and sensor failures. This research introduces TriQXNet, a novel hybrid classical-quantum neural network for Dst forecasting. Our model integrates classical and quantum computing, conformal prediction, and explainable AI (XAI) within a hybrid architecture. To ensure high-quality input data, we developed a comprehensive preprocessing pipeline that included feature selection, normalization, aggregation, and imputation. TriQXNet processes preprocessed solar wind data from NASA's ACE and NOAA's DSCOVR satellites, predicting the Dst index for the current hour and the next, providing vital advance notice to mitigate geomagnetic storm impacts. TriQXNet outperforms 13 state-of-the-art hybrid deep-learning models, achieving a root mean squared error of 9.27 nanoteslas (nT). Rigorous evaluation through 10-fold cross-validated paired t-tests confirmed its superior performance with 95% confidence. Conformal prediction techniques provide quantifiable uncertainty, which is essential for operational decisions, while XAI methods like ShapTime enhance interpretability. Comparative analysis shows TriQXNet's superior forecasting accuracy, setting a new level of expectations for geomagnetic storm prediction and highlighting the potential of classical-quantum hybrid models in space weather forecasting.