Question

Could, quantum mechanically, the Einstein field equations describe a single graviton?

Answer 1

In general, when describing interacting theories at the quantum level, it does not make sense to study isolated particles. Because of self-interactions, a given particle is never truly "als0'. Since the photon does not interact with itself, this issue does not come up (at least when matter is not present). 

2

In the same way photons arise from quantizing electromagnetic waves (which satisfy Maxwell's equations), gravitons should arise from quantizing gravitational waves (which satisfy Einstein's equations). But that's precisely the goal we haven't reached, to build a quantum theory of gravity! This is how Feynman begins his study of GR in his book "Lectures on Gravitation", so I highly recommend reading the first 3 of those lectures if you're really interested

Answer 2

There are various researches showing that the Maxwell equations quantum mechanically describe a single photon. Analogously one may ask, the Einstein field equations also quantum mechanically can describe a single graviton? Sorry if this question doesn't make sense somehow.