Previous studies on hybrid dielectric-graphene metasurfaces have been used to implement induced transparency devices, while exhibiting high Q-factors based on trapped magnetic resonances. Typically, the transparency windows are single wavelength and less appropriate for polarization conversion structures. In this work, a quarter-wave plate based on a hybrid silicon-graphene metasurface with controllable birefringence is numerically designed. The phenomena of trapped magnetic mode resonance and high Q-factors are modulated by inserting graphene between silicon and silica. This results in a broader transmission wavelength in comparison to the all-dielectric structure without graphene. The birefringence tunability is based on the dimensions of silicon and the Fermi energy of graphene. Consequently, a linear-to-circular polarization conversion is achieved at a high degree of 96%, in the near-infrared. Moreover, the polarization state of the scattered light is switchable between right and left hand circular polarizations, based on an external gate biasing voltage. Unlike in plasmonic metasurfaces, these achievements demonstrate an efficient structure that is free from radiative and ohmic losses. Furthermore, the ultrathin thickness and the compactness of the structure are demonstrated as key components in realizing integrable and CMOS compatible photonic sensors.