An array of Ag nanoboxes fabricated by helium-ion lithography is used to demonstrate plasmon-enhanced nonradiative energy transfer in a hybrid quantum well-quantum dot structure. The nonradiative energy transfer, from an InGaN/GaN quantum well to CdSe/ZnS nanocrystal quantum dots embedded in an ~80 nm layer of PMMA, is investigated over a range of carrier densities within the quantum well. The plasmon-enhanced energy transfer efficiency is found to be independent of the carrier density, with an efficiency of 25% reported. The dependence on carrier density is observed to be the same as for conventional nonradiative energy transfer. The plasmon-coupled energy transfer enhances the QD emission by 58%. However, due to photoluminescence quenching effects an overall increase in the QD emission of 16% is observed.