This is a very hypothetical question.
a) we do not have a quantized theory of gravity working in conjunction with elementary particles ( the photon is an elementary particle).
b) we have not detected a graviton in the way we detect photons
If we forget a) and assume for b) that if gravitons exist they are the 0 mass carriers of the gravitational interactions the way the photon is the carrier of the electromagnetic interactions and also assume that what we know of quantum mechanics and kinematics of elementary particles pertains to the graviton too then
no, the photons cannot radiate gravitons and vice versa because both are moving at the velocity of light and cannot be accelerated or decelerated in order to give up radiation in the classical way.
In this hypothesis, as elementary particles, they can interact by scatterings, with a very very low probability due to the smallness of the gravitational interaction. Through higher order diagrams in the scattering crossection, they may change form and produce other elementary particles depending on their energy, or just photons and gravitons of different energies. There will be an equilibrium of sorts.
Without a solid theory of everything all this is speculation.