A viscoelastic model for a shear-deformable microplate is developed in this paper while accounting for geometric nonlinearities. Nonlinear numerical solutions are conducted to examine the resonant oscillations of the microsystem. The geometrically nonlinear theoretical model is developed utilising the Kelvin-Voigt viscoelastic model, to account for nonlinear dissipation, the modified version of the couple-stress theory, to account for small-scale characteristics, and the third-order deformation theory, to account for shear stress. The constitutive relations for both the classical and higher-order stress tensors are constructed and are divided into elastic and viscous components. The elastic components are used to develop the potential strain energy and the viscous components are employed to model the virtual work of damping (energy dissipation). Additionally, the microplate motion energy is developed while accounting for all in-plane, out-of-plane, and rotational motions. A distributed harmonic load, as the representative an external force, is applied to the microsystem and the corresponding virtual work is obtained. The generalised Hamilton's principle is applied to the virtual works and variations of the energy terms, resulting in the nonlinear equations of motion of the microsystem. Being of partial differential type, the equations of motion are discretised into a set of nonlinearly coupled ordinary differential equations consisting of nonlinear geometric and nonlinear damping terms. A solution procedure for the forced oscillation analysis of the microsystem is developed using a continuation method. Different diagrams are constructed to examine the nonlinear resonant characteristics of the viscoelastic shear deformable microplate and to highlight the nonlinear dependency of the Kelvin-Voigt viscoelastic damping mechanism on the oscillation amplitude, for a geometrically nonlinear model.