We theoretically study the magnetic polaron (MP) formation in a ferromagnetic graphene single-electron transistor (SET), which consists of a graphene quantum dot (QD) between two insulating layers, i.e., a ferromagnetic insulator EuO and a nonmagnetic substrate such as SiC. In the lateral direction, the QD is electrically coupled to source, drain and gate electrodes, so that the number of electrons can be controlled by the gate voltage. Due to a proximity effect at the EuO/graphene interface, i.e., the exchange interaction between the charge carriers in graphene and the localized magnetic electrons in EuO, there is a magnetic coupling between the graphene QD and EuO. Using Green's function technique an expression for the total free energy of the SET is derived, which is then minimized at the given temperature and gate voltage. This leads to the MP formation and a consequent local enhancement of the ferromagnetic properties of EuO at the EuO/graphene interface. The MP formation is enforced by the large Coulomb interaction between the carriers in graphene. The spin polarization at the graphene/EuO interface in the QD area can be controlled by the gate voltage of the SET.