We theoretically investigate the properties of second-harmonic generation (SHG) in gold-silicon core-shell nanostructures. We first study a concentric structure. This structure exhibits strong electric field enhancement in the silicon shell due to the combined toroidal dipole mode and electric dipole mode. Efficient SHG can be obtained and the SHG signal is about 5 times as strong as that of the individual Si shell. Further calculations show that the contribution from a surface nonlinear susceptibility at the inner surface of the silicon shell dominates the SHG signal of the core-shell structure. The SHG as a function of wavelength is considered and it shows a resonance behavior. The cases of nonconcentric core-shell structures have also been considered. The SHG is further enhanced in this kind of configuration and the SHG signal can reach about 10 times as strong as that of the concentric case. Our results reveal the strong modification of the SHGs in dielectric nanostructures by using the metal-dielectric hybrid configurations, and could find applications in nanoscale nonlinear devices.