Context. Polarization measurements in the 30─40 GHz range are essential to improving our understanding of Galactic synchrotron radiation and potential anomalous microwave emissions, and obtaining such measurements will ultimately aid in detecting primordial B modes. To achieve this aim, it is crucial to achieve unprecedented performance in instrumentation. Thus, accurately calibrating instrument responses and the polarization angle represents an essential step in state-of-the-art setups. The modeling and mitigation of systematic effects are also critical in this context. Aims. Our objective is to characterize the QUIJOTE Thirty and Forty GHz instrument (TFGI), calibrate it with a reference calibration signal on the ground, compare our results with on-sky calibration based on bright sources, and study the stability of the calibration parameters over time. Methods. First, from the ground, we fit the data using a reference calibration signal (a diode) introduced to resolve degeneracies among the various instrument angles. Finally, we utilized on-sky observations of Tau A and the Moon to validate the results. Results. By creating calibration datasets obtained with the reference diode, we evaluated the data quality and quantified phase switch errors to account for the fine polarization response. We also utilized Tau A and Moon observations to calibrate the system's response and stability over time. In addition, we calculated the refraction index of the Moon to be nMoon = 1.209 ± 0.007 (stat) ± 0.009 (sys) at 31 GHz under smooth-surface assumption. Conclusions. The results from fitting the instrument phase-switch error angle align with 0 deg at 2σ precision, indicating that no further correction is required within a few percent precision. The calibrations with astrophysical sources (Tau A and the Moon) yielded consistent results that constrain the polarization angle and responsivity. The polarization efficiency aligns well with ground measurements and the Tau A characterization, whereas the Moon-based calibration is more affected by systematics. We found hints of responsivity variations over time, although the relative responsivity between channels was found to remain stable. We conclude that installing a live calibrator in the future will enhance the performance of TFGI by continuously monitoring responsivity and in turn improving the mitigation of systematic effects.