The low-energy (intraband) range of the third harmonic generation of graphene in the terahertz regime is governed by the damping terms induced by the interactions. A controlled many-body description of the scattering processes is thus a compelling and desirable requirement. In this paper, using a Kadanoff-Baym approach, we systematically investigate the impact of many-body interaction on the third harmonic generation of graphene, taking elastic impurity scattering as a benchmark example. We predict the onset in the mixed inter- and intraband regime of incoherent features driven by the interaction at four- and five-photon transition frequencies in the third harmonic optical conductivity with a spectral weight proportional to the scattering rate. We also show that in spite of the complex many-body physics, the purely intraband term governing the limit omega -> 0 resembles the constraints of the phenomenological model. We ascribe this agreement to the fulfilling of the conservation laws enforced by the conserving approach. However, the overlap with incoherent features and the impact of many-body-driven multiphoton vertex couplings severely limit the validity of phenomenological description.