Multiband Fano resonances are demonstrated in the near-infrared (near-IR) using an all-dielectric metasurface whose unit cell consists of four silicon nanoblocks on a glass substrate. An in-plane asymmetry triggers symmetry-protected quasi-bound states in the continuum (QBICs), producing multiple high-Q resonances. Their origin is identified through multipolar decomposition of the scattering cross section and field distributions at the resonances. The strong field localization at these resonances enables efficient multiband third-harmonic (TH) generation in the ultraviolet (UV), with a maximum simulated conversion efficiency of 8.5 × 10⁻³ at a peak pump intensity of 1.6 GW/cm². The metasurface is fabricated in symmetric and asymmetric configurations, and its linear and nonlinear responses are measured under normal incidence. A TH conversion efficiency of 1.2 × 10⁻⁶ is obtained at a peak pump intensity of 3.25 GW/cm². These results establish a route to multiband photonic devices, including multiwavelength lasers, multiband harmonic generation, and single-photon sources for quantum photonics.