The structure of electronic energy levels is obtained by solving electronic Dirac equation of graphene quantum disks (GQDs) under the infinite-mass boundary conditions. On this basis, the properties of two-photon absorption (TPA) of GQDs are studied theoretically, and the analytical expression of TPA coefficient associated with electron transitions in the conduction band under an arbitrary size-distribution as well as the two-photon transition selection rule are obtained. The research results show that the peak value of TPA coefficient is about 8 orders of magnitude greater than that of conventional semiconductor quantum dots. The energy spectra and TPA spectra can be adjusted and controlled by the GQDs size, size-distribution function and electron relaxation energy.