We study an independently tunable Fano resonance from quasibound state in the continuum and polarization-independent toroidal dipole resonance in hybrid graphene–dielectric metasurface consisting of a nanoring and a cross-shaped nanobar. The unique structural properties of the proposed metasurface can support directional toroidal dipole mode, where the asymmetric magnetic field enhancement can be dynamically switched on and off while maintaining polarization-independent Fano spectral response. Through reducing or increasing inner radius of nanoring, an additional quasi-BIC dominated by magnetic dipole moment can be excited and independently tuned by altering the polarization direction of the incident wave. In addition, the quasi-BIC can be effectively modulated via adjusting the Fermi energy and layer numbers of the graphene. Our results can be of practical interest for a variety of applications including optical modulator, filter, switches, and light trapping.