Millimeter-wave (mmWave) communication enables high data rates for cellular-connected Unmanned Aerial Vehicles (UAVs). However, a robust beam management remains challenging due to significant path loss and the dynamic mobility of UAVs, which can destabilize the UAV-base station (BS) link. This research presents a GPS-aided deep learning (DL) model that simultaneously predicts current and future optimal beams for UAV mmWave communications, maintaining a Top-1 prediction accuracy exceeding 70% and an average power loss below 0.6 dB across all prediction steps. These outcomes stem from a proposed data set splitting method ensuring balanced label distribution, paired with a GPS preprocessing technique that extracts key positional features, and a DL architecture that maps sequential position data to beam index predictions. The model reduces overhead by approximately 93% (requiring the training of 2 ~ 3 beams instead of 32 beams) with 95% beam prediction accuracy guarantees, and ensures 94% to 96% of predictions exhibit mean power loss not exceeding 1 dB.