General Relativity is a classical field theory, so there's no graviton in it.

However, there are various quantum gravity research programmes which have tried to quantize the (classical) tensor fields of General Relativity in one way or another, for instance by turning (classical) gravitational waves or position and momentum variables into quantum particles. In these theories the graviton is a spin-2 gauge boson and induce an attractive force on fermions with like gravitational charge. This compares with the photon, a spin-1 gauge boson that induces a repulsive force on fermions with like electric charge (and an attractive force on fermions with opposite electric charge). The spin also corresponds with the rank of the (classical) fields, where in General Relativity we have rank-2 tensors and in electrodynamics we have vectors (which are rank-1 tensors).

Unfortunately this approach has a problem: particles other than fermions also have a gravitational charge, including gravitons themselves. (Photons do not have an electric charge, but they do have a gravitational charge). Quantum field theories use a variety of tricks to remain linear theories, but General Relativity's Einstein Field Equations are non-linear. That non-linearity manifests in graviton-graviton interactions in this sort of quantized General Relativity, and we do not know how to stabilize the resulting quantum field theories.

In more concrete terms, let's "drop" an electron from higher potential to lower potential. The electron is a fermion with electric and gravitational charge. When we drop it from higher to lower electric potential, out pops a photon. When we drop it from higher to lower gravitational potential, out pops a graviton. But we can't move the photon around the electric field and get it to produce another photon, because the photon is not self-interacting. A graviton, on the other hand, can be dropped down a gravitational potential causing it to emit another graviton. That in turn influences the potential, making it easier for those two gravitons to each emit a further graviton, which influenes the potential, making it easier for those ... and so forth to infinity. Our original electron is in that potential too, so it in turn can emit further gravitons as well. Fortunately this explosion of gravitons only happens in strong gravity, and we only find that deep inside black holes (or in the very early universe), rather than somewhere we can outright observe (assuming the cosmic censorship principle is true), so we can still use this type of theory to describe most of the universe.

Unfortunately, gravitational waves are hard enough to detect; when they are quantized, they turn into lots and lots and lots of gravitons, each of which has a tiny fraction of the detectability of a gravitational wave. Detecting an individual graviton experimentally is beyond the ability of present human technology, or anything reasonably likely to appear in the next few decades. So if we ever figure out how to stabilize this type of approach to quantum gravity, we will only be able to assess it against observations of many-graviton systems, likely at extreme events like black hole/neutron star mergers or asymmetrical hypernovae.

This is just one broad example of quantum gravity; there are several other approaches, some with a "graviton" dynamics (e.g. string theory's gravitons are a string state) and some without.

All these approaches share a common problem: General Relativity describes our universe exceptionally well in practice, and its fundamental components are among the best-tested of any scientific theory to date. Where there is weak gravity, barring significant masses in superpositions of position, any type of quantum gravity has to exactly reproduce the results of General Relativity.

It is likely that we will be able to bring microgram masses into such superpositions in the next few years, and we will be able to directly probe one aspect of quantum field theory on (classical) curved spacetime (aka semiclassical gravity). However, if General Relativity fails in that situation, it would certainly allow some useful comparisons between competing theories (with and without gravitons) hoping to extend General Relativity as it in turn extended Newtonian gravity. If only graviton-containing theories match the experimental results, that would be evidence in favour of the existence of a graviton, even without direct detection; conversely, perhaps only non-graviton theories match, in which case the existence of a graviton would seem pretty dubious.

Right now, though, we just don't know: in the absence of evidence for or against, gravitons are purely theoretical.