Abstract To get a carbon-based qubit, we pay attention to the two-electron conduction band of a graphene quantum dot (GQD) in the presence of an external magnetic field and an extrinsic Rashba spin-orbit interaction (SOI). To help understand the formation of the two-electron spectra, we first calculate the tight-binding (TB) spectra. There exist the sensitivity of the conduction band to magnetic fields and the mixing of spin states induced by a Rashba SOI. The two factors inspire the study of the magnetic-field modulation of the conduction band for realizing a spin qubit. We present the method for calculating the electronic structure of a few-electron GQD. The roles of the Coulomb interaction and the Rashba SOI in the two-electron conduction band are investigated. The Coulomb interaction contributes to a singlet-triplet level crossing and the Rashba SOI leads to a singlet-triplet mixing. The fast initialization and coherent manipulation of spin states are demonstrated by the magnetic control of singlet-triplet splitting.