Plasmonic nanoparticle dimers with interparticle gap distances (d) in the nanometer scale are able to produce huge electromagnetic field enhancements in the gap region, useful for novel optical applications. In this study, using time dependent density functional theory calculations, we present a comprehensive study on the evolution of the plasmon coupling of the plasmonic heterodimers consisting of closely spaced nanospheres and nanocubes: “Na nanocube & Na nanosphere”, “Mg nanocube & Mg nanosphere”, “Mg nanocube & Na nanosphere”, and “Na nanocube & Mg nanosphere”. We found that the individual Mg and Na nanocubes and nanospheres exhibit localized surface plasmon resonances (LSPRs). Importantly, the heterodimer formation causes highly interesting developments in the plasmon coupling. We believe that the present theoretical findings will help in the design of plasmonic heterodimers made from Earth-abundant metals.