In this study, an all-dielectric metasurface refractive-index sensor consisting of an array of cylindrical nanoholes is proposed. In-plane symmetry breaking of the structural unit is introduced by translating the nanoholes, and asymmetric dielectric nanohole arrays can achieve quasi-bound states in the continuum (BIC) resonance mode with a high-quality factor (Q-factor) by electric quadrupole (EQ). The relationship between the structural asymmetry parameters and the radiative Q-factor of the Fano resonance is theoretically analyzed. It is found that the mode is a symmetry-protected BIC, and near-field analysis and multipole decomposition show that the EQ plays a dominant role in the resonance mode. In addition, the influence of structural parameters on Fano resonance is analyzed, and the spectral response of the refractive index of the medium is calculated. The sensitivity of the structure reaches up to 512 nm/RIU (refractive index unit), the Q-value is 2568.7, and the figure of merit (FOM) is 760. The proposed structure has potential applications in highly sensitive biosensors in the near-infrared range.