Seeking operative terahertz (THz) devices has always stimulated considerable attention. Of particular interest is the THz beam splitter. Here, a tunable THz polarizing beam splitter (PBS) is proposed based on a graphene-embedded quarter-wave stack with a central defect layer of air. The spectral performance of the structure is investigated by the transfer matrix method. It is found that the electromagnetic waves can be decomposed into two separate polarized waves at incident angles greater than the critical angle. Furthermore, it is shown that a new kind of Brewster angle is found at the low THz frequencies due to the existence of the graphene nano-layers. The appearance of this angle which we call it the graphene induced Brewster angle results in the separation of TM-and TE-polarized waves at the low THz frequencies (f < 2 THz) in addition to the high THz region. It is shown that the working frequency range of PBS can be easily tuned by adjusting the width of the defect layer of air and also by tuning the chemical potential of graphene nano-layers via a gate voltage. The analysis of the proposed PBS is confirmed by the Finite Element Method (FEM) simulations that were performed with the commercial software COMSOL 5.2. Our investigations also reveal that the high transmission extinction ratio (> 200 dB) for TM waves with frequency f < 2 THz is achieved by increasing the chemical potential to 0.5 eV. Moreover, this structure can exhibit extremely high extinction ratio for TE waves with high THz frequencies. Finally, the degree of polarization equals to one is reported for the PBS proposed here. This structure offers the opportunity to realize a high-efficiency PBS with very high extinction ratios at the broadband THz frequency.