The emergence of new materials will drive the development of a generation of devices. The purpose of this study is to realize the nonlinear optical frequency conversion in the novel material structure - metasurface structures. We study the second harmonic generation (SHG) from the all-dielectric resonant meta-grating metasurfaces with high quality (Q) factors. The quasi-bound states in the continuum (Q-BIC) modes can be easily induced using the linearly polarized plane wave with normal incidence. Two types of asymmetric one-dimensional meta-grating metasurfaces with square air notch defect and different widths based on AlGaAs are numerically investigated. When the symmetry of the AlGaAs meta-grating structures is broken, the BIC modes are transformed into the Q-BIC modes, which support strong local field confinement in the AlGaAs meta-grating array, thereby promoting the SHG conversion. When the pump laser intensity is about 5 MW/cm2, the two type structures provide the SHG conversion efficiencies about 0.059 and 0.166. The investigation results will promote the application of nonlinear metasurfaces in integrated nonlinear optical systems.