We present an experimental study of optical second harmonic generation (SHG) from arrays of nanostructures exhibiting collective plasmon resonances in the visible spectral range. Gold nanoparticles with the lateral diameter of 100 nm are packed in a square lattice with the period of 400 nm and are covered by a 90 nm thick iron garnet layer. We show an enhancement of SHG in the spectral vicinity of the lattice surface plasmon resonance. It is demonstrated that application of the external DC magnetic field results in a spectral shift of the SHG maximum up to 5 nm, accompanied by a pronounced phase modulation of the second harmonic wave. This spectral shift significantly prevails over the analogous linear magneto-optical effect and is explained by the interference of resonant and nonresonant SHG contributions of various nature.