In this work, we performed theoretical calculations based on density functional theory for graphene quantum dots considering three different sizes. We consider graphene quantum dots formed by 7, 19, and 37 rings of C atoms in a hexagonal arrangement. The electronic band structure of graphite and graphene were calculated to evaluate the methodology and parameters used during calculations. The modeled graphene quantum dots structures were initially passivated by H atoms, then the H atoms were gradually replaced by OH or COOH radicals to investigate the influence of oxygen on the chemical stability and the energy gap. Based on the electric dipole moment, the replacement positions were selected. The results demonstrate that all the structures are chemically stable and that the energy gap depends on the number of OH or COOH radicals at the edge of the GQDs. H passivation results in energy gaps of 2.83 eV, 1.87 eV and 1.33 eV, decreasing with increasing OH or COOH radical amount. It was found that the energy gap varies non-monotonically as the OH or COOH radicals increase. To better understand the origin of the energy gap and its changes by OH or COOH radicals, we calculated the DOS and evaluated the HOMO and LUMO by Fermi surfaces.