In this thesis, we aim to study magnetic properties of hexagonal shaped graphene quantum dots with armchair edge in the case of atomic collapse by modelling two vacancies on it. The measured relativistic electron transport property of the graphene allows us to observe the phenomenon called "atomic collapse" in a small energy scale which existence is proven theoretically before for atoms whose atomic number is higher than 170. First we modelled a Coulomb potential at the center of a hexagonal shaped and armchair edged GQD and examined by using tight-binding method. We obtain similar results with previous works. After that, we started to study magnetic properties of the dot by meanfield Hubbard method which includes spins into calculation. We modelled a vacancy close to the center of the dot and examined electronic and magnetic properties by MFH metod. Also we modelled two vacancies on the dot that we changed the distance between them and the direction respectively. Also by applying Coulomb potential at the center of the vacancies we examined magnetic behaviour at the atomic collapse regime. Also, we compared our results with the works obtained by using RKKY (Ruderman-Kittel- Kasuya-Yosida) interaction method which considers the indirect interactions of magnetic impurities that uses electrons of metallic substrates. We found that increasing Coulomb potential and increasing distance between the vacancies, reduces correlations of electrons around the vacancies. The ground state energy difference between ferromagnetic and antiferromagnetic systems, that proportional to interaction strength, shows similar behaviour that has been observed by using RKKY method. Also if we take out two atoms from the same sublattice and with the same spin property, changing Coulomb potential leads to ferromagnetic-anti-ferromagnetic phase transition, independent from the atomic collapse behaviour. Also we observed that there is no direct link between the magnetic transition and the energy difference of the vacancy states.