Developing the broadband controllable or tunable terahertz (THz) polarization and phase devices are in an urgent need. In this paper, we demonstrate a broadband controllable THz quarter-wave plate (QWP) with double layers of graphene grating and a layer of liquid crystals. The double layer graphene gratings can achieve a switchable QWP to switch between linear-to-linear and linear-to-circular polarization states with over 0.35THz bandwidth in the ON or OFF state by applying biased electric field on the graphene grating or not. Moreover, this QWP based on the structure of periodic gradient grating can significantly enhance the phase difference between two orthogonally polarized components compared to that based on equal-periodic grating structure because of the additional phase distribution of the gradient structures. Furthermore, we incorporate liquid crystals into the graphene grating to form a tunable QWP, of which operating frequency can be continuously tuned in a wide frequency range by electrically controlling the molecular director of the liquid crystals. The results show that the graphene periodic gradient grating with LCs not only broadens the operating bandwidth, but also reduces the external electric field. This device offers a further step in the development of THz polarization and phase devices for potential applications in THz polarized imaging, spectroscopy, and communication.