Hybrid metal-dielectric nanostructures are highly effective and helpful in generating and enhancing linear and nonlinear optical phenomena due to their unique optical properties. In this study, the linear optical responses, and second- and third-order harmonics have been calculated numerically for this structure. Nonlinear optical response efficiency has been studied at the resonance wavelengths of the nanostructure by varying its parameters (radius and length). The desired efficiency can be achieved by adjusting the radius and length of the nanorods and matching the resonance wavelengths with the fundamental and generation frequencies of nonlinear optical phenomena in specific applications. Moreover, the effect of the number of gold nanorods on harmonic generation has been investigated. By increasing the number of nanorods in pairs, it is possible to generate high-order harmonics. The acquired results can be used to improve optical devices such as biomedical equipment, optical integrated circuits, and sensors.