The bending analysis of a viscoelastic double nanoplate system is performed. The analysis is based on the modified couple stress theory and relaxation theory. An analytical solution for the transverse relative deflection of two nanoplates is obtained with help of Navier's method. The influence of geometrical and physical factors on the bending behavior of the nanoplates is discussed. It was shown that the transverse relative deflection of two nanoplates depends upon the relaxation time, ratio of the delayed to initial extensional elastic moduli, and damping parameters. A decrease in the material length parameter causes a steady increase in the transverse relative deflection. The structural damping of the viscoelastic nanoplate and the viscoelastic medium between two nanoplates leads to a decrease in the transverse relative deflection.