Nanolaminate metamaterials recently attracted a lot of attention as a novel second-order nonlinear material that can be used in integrated photonic circuits. Here, we explore theoretically and numerically the opportunity to enhance the nonlinear response from such nanolaminates by exploiting Fano resonances supported in grating-coupled waveguides. The enhancement factor of the radiated second harmonic signal compared to a flat nanolaminate can reach values as large as 35 for gold gratings and even 7000 for MgF2 gratings. For the MgF2 grating, extremely high-Q Fano resonances are excited in such all-dielectric system that result in strong local fields in the nonlinear waveguide layer to boost the nonlinear conversion. A significant portion of the nonlinear signal is also strongly coupled to a dark waveguide mode, which remains guided in the nanolaminate. The strong excitation of a dark mode at the second harmonic frequency provides a viable method for utilizing second-order nonlinearities for light generation and manipulation in integrated photonic circuits.