We experimentally demonstrate for the first time the existence of distinguishable contributions to hyper-Rayleigh scattering (HRS) intensities from Ag nanoparticles arising from electric-dipole and electric-quadrupole plasmon resonances at the emitted wavelength. We show that these results can be successfully modeled using an electromagnetic theory of HRS which assumes a surface-induced nonlinear susceptibility. In addition, we show that simple angular distribution measurements may be used to determine the relative sizes of the dipole and quadrupole contributions.