We consider a nanosystem consisting of two coplanar uniformly charged nanodisks that are coupled via Coulomb forces. Such a model represents a typical situation encountered in two-dimensional semiconductor quantum dot systems of electrons. We provide an exact integral expression for the interaction energy between the two coplanar nanodisks as a function of their separation distance. It is found that the difference between a standard Coulomb potential and the current one has features reminiscent of a Lennard-Jones interaction potential. The results derived can be useful to understand formation of clusters and/or aggregates in systems of coplanar charged nanodisks that contain electrons.