Global Navigation Satellite Systems (GNSS) are critical for Positioning, Navigation, and Timing (PNT) applications. However, GNSS are highly vulnerable to spoofing attacks, where adversaries transmit counterfeit signals to mislead receivers. Such attacks can lead to severe consequences, including misdirected navigation, compromised data integrity, and operational disruptions. Most existing spoofing detection methods depend on supervised learning techniques and struggle to detect novel, evolved, and unseen attacks. To overcome this limitation, we develop a zero-day spoofing detection method using a Hybrid Quantum-Classical Autoencoder (HQC-AE), trained solely on authentic GNSS signals without exposure to spoofed data. By leveraging features extracted during the tracking stage, our method enables proactive detection before PNT solutions are computed. We focus on spoofing detection in static GNSS receivers, which are particularly susceptible to time-push spoofing attacks, where attackers manipulate timing information to induce incorrect time computations at the receiver. We evaluate our model against different unseen time-push spoofing attack scenarios: simplistic, intermediate, and sophisticated. Our analysis demonstrates that the HQC-AE consistently outperforms its classical counterpart, traditional supervised learning-based models, and existing unsupervised learning-based methods in detecting zero-day, unseen GNSS time-push spoofing attacks, achieving an average detection accuracy of 97.71% with an average false negative rate of 0.62% (when an attack occurs but is not detected). For sophisticated spoofing attacks, the HQC-AE attains an accuracy of 98.23% with a false negative rate of 1.85%. These findings highlight the effectiveness of our method in proactively detecting zero-day GNSS time-push spoofing attacks across various stationary GNSS receiver platforms.