Optical second-harmonic generation (SHG) is considered as a technique for probing quantum well states in ultrathin noble metal films deposited on Si. Due to the absence of screening effects in SHG the electronic resonances an clearly resolved by the non-linear technique in contrast to linear optical spectroscopy. The physical origin of SHG in quantum well samples is discussed in terms of a microscopic model. From numerical calculations and an investigation of linear reflectance data it is argued that the spatial variation of the pump field driving the non-linear response may be described using an isotropic Drude model of the dielectric properties. This allows us to interpret the experimental data by comparison to a relatively simple one-dimensional model. The calculated and measured thickness dependencies of the second-harmonic signal are in excellent agreement and both position and intensity of the electronic resonances are reproduced by theory.