In this paper, an all-dielectric metasurface consisting of a unit cell containing a nanocube array and organized periodically on a silicon dioxide substrate is designed and analyzed. By introducing asymmetric parameters that can excite the quasi-bound states in the continuum, three Fano resonances with high Q-factor and high modulation depth may be produced in the near-infrared range. Three Fano resonance peaks are excited by magnetic dipole and toroidal dipole, respectively, in conjunction with the distributive features of electromagnetism. The simulation results indicate that the discussed structure can be utilized as a refractive index sensor with a sensitivity of around 434 nm/RIU, a maximum Q factor of 3327, and a modulation depth equal to 100%. The proposed structure has been designed and experimentally investigated, and its maximum sensitivity is 227 nm/RIU. At the same time, the modulation depth of the resonance peak at λ = 1185.81 nm is nearly 100% when the polarization angle of the incident light is 0 °. Therefore, the suggested metasurface has applications in optical switches, nonlinear optics, and biological sensors.