In this Letter, we propose that the photopatterned liquid crystal (LC) can act as a broadband and efficient terahertz (THz) Bessel vortex beam (BVB) generator. The mechanism lies in the frequency-independent geometric phase modulation induced by the spatially variant LC directors. By adopting large birefringence LCs and optimizing the cell gap, the maximized mode conversion efficiency can be continuously adjusted in broadband. Furthermore, the LC patterns can be designed and fabricated at will, which enables the THz BVBs to carry various topological charges. Such a THz LC BVB generator may facilitate the advanced THz imaging and communication apparatus.