In this Letter, we report a prototype system that realizes the complete functionality for quantum two-way time transfer, comparison, and synchronization between two integrated terminals. The synchronization performance was tested over a 50-km spooled fiber link. With the common frequency reference, the time deviation was measured as 0.45 ps at an averaging time of 104 s, which is limited by the system's hardware and determines the minimum achievable synchronization stability. By employing an open-loop fiber-optic microwave frequency transfer in combination with the technique of dynamically identifying and steering the time offset between the terminals, a synchronization stability of 1.26 ps at 104 s was achieved. Further utilizing the grey prediction model to correct the time offset data, the synchronization stability was significantly improved to 0.69 ps at 104 s, showing its potential to enhance the synchronization performance. This report marks the development of a utility quantum two-way clock synchronization system. The ongoing exploration of advanced time-offset adjustment strategies to attain synchronization stability significantly below 1 ps is poised to yield invaluable benefits for future applications.