Graphene-based materials have been proposed as promising electrodes for electric double layer capacitors. Recently, it has been found that one of the limitations of graphene electrodes is the finite quantum capacitance at low applied voltage. In this work, we investigate the impact of having point-like topological defects in graphene on the electronic structure and quantum capacitance. Our results clearly show that the presence of defects, such as Stone Wales, di-vacancies, and di-interstitials, can substantially enhance the quantum capacitance when compared to pristine graphene, which is found to be due to defect-induced quasi-localized states near the Fermi level. In addition, the charging behavior tends to be asymmetric around the neutrality point. We also discuss the possibility of tuning the electronic structure and capacitance through mixtures of these defects. Our findings suggest that graphene-based electrodes with topological defects may demonstrate noteworthy capacitance but should be carefully selected for use as either the positive or negative electrode.