The graviton

[QuantumT]

Where is the graviton expected to be located in the nucleus? I know that up and down quarks have a spin of ½, and the graviton's should be 2. Could it be inside the quarks? Or could it be an undetectable sphere around the nucleus as a whole?

[Janus]

Gravitons would be quanta of gravitational radiation( gravitational waves).  You would not "find them" anywhere in the nucleus.  They would be produced under the type of conditions that generate gravitational waves, in the same way that photons are produced by the generation of electromagnetic radiation ( radio waves, light, etc)

[Eise]

Why would you expect the (hypothetical!) graviton at some place? Photons are also not located anywhere; if a photon exists, it moves, necessarily with the speed of light. As the graviton, if it exists, is also massless, it also will travel at the speed of light. So it cannot be 'located' anywhere before it starts its journey. For gravitons and photons holds: 'to be is to move'.

[MigL]

And quarks are a fundamental Fermionic particle ( like electrons and neutrinos ), so they don't 'contain' any other particles.
Although they can interact with certain Bosonic particles ( such as electron with photon  and quarks with gluons ).

[joigus]

If you mean real gravitons, "free-flying gravitons" so to speak, that's out of the question. They should be what gravitational waves are made of, and GW are difficult enough to detect themselves. Let alone the quanta that (presumably) make them up.

If you mean virtual gravitons, no virtual particle can be detected. They are un-physical or "off-shell" (they don't satisfy Einstein's mass-energy relation).

For gravitons to be detected you would have to scatter them with massive particles* or other gravitons, and the cross section is so small, due to the smallness of the coupling at any reasonable energy that we wouldn't see anything.

Besides, as Eise has pointed out, messenger particles, like photons, don't really have localization in the normal sense. This is a consequence of quantum field theory. In QFT, you cannot define a position operator. Because massive particles can move in a non-relativistic regime, you can approximately define position for them if they're in the non-relativistic regime. But photons and other gauge bosons cannot be considered as non-relativistic (except possibly the Z and W⁺, W^- before they decay), so they cannot given position in any precise sense in QFT.

*Actually, with anything that has energy.

Edited July 20, 2020 by joigus
Added footnote.

[Mordred]

If a hypothetical graviton is found. (Though we would need far higher energy levels at an LHC) it would most likely be a spin two boson.

 We're nowhere near the technology to produce one (TeV ) energy range. Wish we could though quantizing gravity would be easily done.