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Dielectric metasurfaces play an increasingly important role in enhancing optical nonlinear generations owing to their ability to support strong light-matter interactions based on Mie-type multipolar resonances. Compared to metasurfaces composed of the periodic arrangement of nanoparticles, inverse, so-called, membrane metasurfaces offer unique possibilities for supporting multipolar resonances, while maintaining small unit cell size, large mode volume and high field enhancement for enhancing nonlinear frequency conversion. Here, we theoretically and experimentally investigate the formation of bound states in the continuum (BICs) from silicon dimer-hole membrane metasurfaces. We demonstrate that our BIC-formed resonance features a strong and tailorable electric near-field confinement inside the silicon membrane films. Furthermore, we show that by tuning the gap between the holes, one can open a leaky channel to transform these regular BICs into quasi-BICs, which can be excited directly under normal plane wave incidence. To prove the capabilities of such metasurfaces, we demonstrate the conversion of an infrared image to the visible range, based on the Third-harmonic generation (THG) process with the resonant membrane metasurfaces. Our results suggest a new paradigm for realising efficient nonlinear photonics metadevices and hold promise for extending the applications of nonlinear structuring surfaces to new types of all-optical near-infrared imaging technologies.