Everything which has mass has gravity. Some theories hypothesis that gravitons are responsible for that, but do they have any clue where they come from? Like gluons are interacting between quarks and photons between eg electrons. But does gravitons have, according to those theories, also have a kind of particle on their own with whom they are interacting with. Or is it possible that fe gravitons are also coming from quarks?
There is no fully satisfactory quantum theory of gravity. The electromagnetic field is quantized, the quantum excitations are photons. It would be reasonable to believe that the gravitational field would also be quantized, and the quanta would be particles, called gravitons. In fact it would be very hard to explain how gravity could not be quantised.
The image of a particle emitting photons is rather misleading. Instead, two particles exchange a photon in an electromagnetic interaction. And the particles are not classical, but quantum mechanical.
Photons carry the electromagnetic force between any charged particles: not just electrons, but also muons, tauons, quarks, and the W particles. By this analogy, anything that can create a gravitational field would be exchanging gravitons: that includes anything that has mass or momentum. (ie more or less anything).
However, there is neither a theoretical, nor an experimental basis for the existence of gravitons.