Where do gravitons reside - in atoms? As a non-physicist, I imagine that if we have an atom of lead and an atom of copper floating near each other in deep space with no initial relative velocities, they will exert a gravitational pull on each other and accelerate towards each other.
Am I right? If so, where are the gravitons that cause this? Are there some in each atom? If I'm wrong, what is the true state of affairs as far as we know it?

P.S. If gravity is not a force, what is the supposed function of a graviton? Does it warp space around an atom?
 A: 
As a non-physicist, I imagine that if we have an atom of lead and an atom of copper floating near to each other in deep space with no initial relative velocities, they will exert a gravitational pull on each other and accelerate towards each other.

Yes this is right.

Am I right? If so, where are the gravitons that cause this? Are there some in each atom? If I'm wrong, what is the true state of affairs as far as we know it?

There are lots of different ways of talking about any force, and in particular gravity, that seem mutually incompatible but are all capturing different aspects of the same underlying reality. I'll get to your question about gravitons, but I don't think what I'll say is particularly useful, so I'll give you some other ways to think about it first that I think may be more helpful.

*

*For weak gravitational fields (such as the one produced by an atom), we can simply think of gravity as a force as first described by Newton. The two atoms have mass, and so exert an (utterly, ridiculously small) attractive force on each other.

*In General Relativity -- which is not a quantum mechanical theory -- we can think of mass as curving spacetime, and what appears to us to be a gravitational force is really the motion of objects through a curved spacetime. It's also completely fine to use this picture, and say that the atoms produce an (utterly, ridiculously small) amount of spacetime curvature, and each atom moves in the curvature produced by the other atom (and itself).

*We don't have a full quantum theory of gravity. But we do have a perturbative description, which works when the gravitational fields are weak. In this case, the gravitational attraction can be described as an exchange of virtual gravitons between the atoms, in the same way that electromagnetic attraction between a proton and electron can be described by the exchange of virtual photons. Your question "where are the gravitions before the interaction" can also be asked of photons. The answer is that this is a misunderstanding of what is going on. You can think of graviton is a small fluctuation in a gravitational field that permeates all space and is always there (you can also think of a photon as a small fluctuation in an electromagnetic field that permeates all space and is always there). The graviton is created randomly by one atom, and destroyed randomly by the other atom (aside: more precisely there is a probability amplitude for this process to occur and we need to sum the probability amplitudes for all possible processes). As an analogy, you can think of a violin string. If I pluck the string, I might hear the note C. Where was the C note before I plucked the string? I would say it's not there -- the string is always there, but the note is only present when I pluck the string, and eventually the note dissipates and disappears.

I just want to emphasize that even though the words I used in the above 3 descriptions are different, they all make practically indistinguishable predictions for the gravitational attraction of atoms. That's all physics can really do in the end -- make predictions about the outcomes of experiments. A popular hypothesis is that one day we will find a deeper explanation that supersedes all these different ways of talking about gravity, and which reduces to them in various limits. But, we don't know if this explanation exists, and we don't know with great confidence what it is.

In the comments, @CosmosZachos made a good point, which I summarize here. The number of photons, or gravitons, is not a conserved quantity. For example, an electron and positron can annihilate and produce two photons. This may be unlike your intuition of "ordinary matter", like in chemistry, where we expect the number of atoms of a given element to be conserved in a "typical" chemical reaction (not involving nuclear reactions). But there is no problem here -- energy, momentum, electric charge, and angular momentum are all conserved, and there is simply no requirement that the number of photons or gravitons is conserved.
A: In quantum physics, gravitational force is exerted by two massive particles emitting gravitons and receiving each other's gravitons. The gravitons exist only while in transit between the two masses, and travel at (or very close to) the speed of light.
The graviton is one of several force-carrying particles, collectively known as bosons. The others include the photon which carries the electromagnetic force, mesons which carry the weak nuclear force and gluons which carry the strong nuclear force.
But calling any of these objects a "particle" is misleading. Technically, a graviton is a wave-like excitation of the zero-point graviton field between the two massive objects and its propagation is governed by a wave equation. The wave "collapses" when the graviton is absorbed.
But in general relativity, the graviton represents a tiny, quantised curvature or kink in spacetime. A big pulse of gravitons can warp spacetime enough for us to detect the event by watching how it affects laser beams. Enough gravitons packed close together can comprise such a sharp curvature that spacetime collapses in on itself and creates a black hole.
Mathematically, the equations of general relativity describe spacetime curvature as a macroscopic geometry, while the equations of quantum field theory describe quantum field excitations as waves in some field which permeates that spacetime. The two kinds of equation have some tantalising similarities (especially for electromagnetic photons), but there are also major inconsistencies between the two approaches.
One problem is how a quantum gravity field can overlay a background spacetime when it is itself part of the fabric of relativistic spacetime.
Another problem hinges on the time-reversibility of the quantum equations. If you run them backwards in time they still make sense. But some events in relativity are irreversible, such as when a black hole forms. Black holes do eventually evaporate, but this is through a quantum mechanism (Hawking radiation) not a relativistic one.
Finding any unifying mathematical or conceptual framework has proved extremely elusive. A Nobel prize awaits whoever can come up with one!
A: The difference between photons and gravitons is in their probabilities to be emitted. Emmitting gravitons is highly improbable.
But if one speaks of the attraction force, it is always here, (Newton law $F\propto G\cdot m_1 m_2/r^2$). Nevertheless, in atomic problems this force is still too small, and it never is taken into account.
