The nonlinear vibration of sandwich viscoelastic plates under wide-band random excitation is investigated. The main attention is put on the influence of the one-to-one internal resonance, arisen from the close natural frequencies of the asymmetric modes of a near-square plate, on the response. The multi-modal response and the on-off intermittency phenomenon are especially considered. The mathematical modeling of the mid-layer is based on the moderate transverse shear strains and rotations, which have led to both geometrical and material nonlinearities. For the nonlinear constitutive equation of the mid layer, a single integral viscoelastic model is used. The displacement field in the thickness direction is also assumed to be linear for the in-plane components and quadratic for the out-of-plane components. Moreover, the Kirchhoff theory with the von-Karman nonlinearities are used for the outer layers. The solution is initiated by applying the perturbation method along with the Galerkin’s method to obtain integro-differential ordinary equations in time. These equations are then, solved using the Gaussian and non-Gaussian closure methods and the results are used to investigate the occurrence of the bifurcation with the aid of the Pseudo-arclength continuation method. Numerical results are presented for the multi-modal response and the minimum excitation intensity required for the nonlinear interaction between asymmetric modes.