Weak hyperfine interaction as well as spin-orbit interaction enforced that graphene quantum dot qubit material rather than other semiconductor qubits material. So; we suggest here a Hamiltonian model of a graphene nanoribbon double quantum dot qubit based on the functionalization of graphene nanoribbon by hydrogen atoms to produce a theoretical studied on a quantum computer. We are using the Dirac equation and the Heisenberg exchange approach to solve our model. Then we determine the exchange interaction J_ex. We investigate the effects of potential barrier height and barrier thickness d on exchange coupling J_ex. Our results show a great variation of J_ex depends on these parameters, and how this parameter affected on J_ex at special value. Also, we can use the variation of J_ex with both potential barrier height and barrier thickness d to represent how the information can transfer, which is important to gate operation necessary for quantum information.