Asymmetric metasurfaces supporting quasi-bound states in the continuum (-BICs) have recently attracted significant interest in the field of nanophotonics due to their high quality factor and strong light-matter interaction properties. However, asymmetric metasurface structures are susceptible to the polarization state of the incident light, which constrains their potential applications. In this Letter, we present a new, to our knowledge, scheme of polarization-independent quasi-BIC resonance supported by a non-rotationally symmetric nanorod dimer metasurface. By tuning the asymmetry parameter, the designed metasurface exhibits a consistent quasi-BIC response for incident plane waves of arbitrary polarization. The physical mechanism of the quasi-BIC resonance is elucidated by the study of the far-field multipole decomposition and the near-field electromagnetic distribution. We then point out that the realization of the polarization-independent quasi-BIC resonance depends on the transition between magnetic and electric quadrupoles. Furthermore, the designed metasurface is demonstrated to have excellent refractive index sensing performance. This work provides a new idea for the design of polarization-independent and high-performance resonators.