The coupling between excitonic and plasmonic nanoparticles produces collective optical properties, which are quite different from the properties of the individual components, leading to modification of the optical properties, such as the emission, the dispersion and the absorption, of hybrid quantum dot-plasmonic nanoparticle structures. During the last years, these interesting optical effects have attracted the scientific interest, both on an experimental, as well as, on a theoretical level, in hybrid nanostructures which are composed of semiconductor quantum dots and metal nanoparticles. The study of the Λ-type system that describes the asymmetric double- semiconductor quantum dot system (asymmetric quantum dot molecule) has also attracted the scientific interest of several scientists, who investigated various effects, including the pump-probe response and the four-wave mixing, as well as, the Autler-Townes splitting and the tunneling-induced transparency effects. In this study, the four-wave mixing spectrum of a strongly pumped hybrid structure is theoretically examined. The hybrid structure consists of an asymmetric double semiconductor quantum dot molecule and a spherical metal nanoparticle, which are coupled together via long-range Coulomb interaction. Having as a starting point the Hamiltonian of the system, in the dipole and the rotating-wave approximations, we derive a set of nonlinear density matrix equations, which are numerically solved, in the steady-state limit, and then the four-wave mixing coefficient is calculated within a range of values of the pump-probe field detuning. The spectral response of the four-wave mixing coefficient is investigated, for different values of the pump-field detuning, the electron-tunneling rate and the energy gap between the upper states in the energy-level scheme of the double semiconductor quantum dot molecule, while the interparticle distance between the two components of the structure is modified.