We propose a controllable qubit in a graphene nanobubble (NB) with emergent two-level systems (TLSs) induced by pseudo-magnetic fields (PMFs). We found that double quantum dots can be created by the strain-induced PMFs of a NB, and also that their quantum states can be manipulated by either local gate potentials or the PMFs. Graphene qubits clearly exhibit avoided crossing behavior as electrical detuning, with energy splittings of about few meV. We show a remarkable tunability of our device design that allows a fine control of the Landau-Zener transition probability by strain engineering of the NB, showing half-and-half splitting at the avoided crossing point. Further, we demonstrate that the TLSs in the NB exhibit Rabi oscillations near the avoided crossing point, resulting in very fast Rabi cycles of a few ps.