The transmission properties of a one-dimensional graphene-based Thue–Morse quasiperiodic structure in which graphene nanolayers are embedded between dissimilar adjacent dielectric layers have been investigated. From the numerical results performed by transfer matrix method, it is found that the structure possesses an omnidirectional photonic bandgap called a graphene-induced photonic bandgap (GIPBG) due to the existence of the graphene nanolayers, in addition to the structural Bragg gaps. To design an omnidirectional broadband THz filter, the effects of many parameters such as the scale length and relative permittivity of the layers on the properties of GIPBG are discussed. Our findings show that the frequency range of the omnidirectional GIPBG increases as the scaling decreases. In contrast to Bragg gaps, the width of such an omnidirectional bandgap is notably enhanced by reducing the contrast of the layers’ permittivity and reaches a maximum value when the constituent layers have the same relative permittivity. Our investigations also reveal that in the case of a uniform material, the lowest permittivity of the dielectric material would result in the widest omnidirectional GIPBG. Moreover, the possibility of external control of the omnidirectional bandwidth using a gate voltage is shown. Finally, the application of the structure as a polarizing beam splitter has been shown.