A terahertz (THz) q-plate is proposed and demonstrated to generate THz vortex beams. It is composed of a large birefringence liquid crystal (LC) with spatially-varying director distribution sandwiched by two pieces of fused silica glass. A polarization-sensitive alignment agent is photopatterned to carry out the specific LC director distribution. THz vortex beams with different topological charges are characterized with a THz digital holographic imaging system. The intensity and phase distributions consistent with theoretical analyses are obtained. Besides, an eight-lobed intensity distribution is observed corresponding to the vertical polarization component of a cylindrical vector beam. This work may inspire novel THz applications.