Could the curvature of spacetime, as in general relativity, result from the interaction of quantum fields? If both the general and special theories of relativity deal with space as spacetime, then the special theory of relativity deals with spacetime as flat, and the general theory of relativity deals with it as a curvature in the presence of the forms of matter, energy, and quantum. Field theory treats space-time as flat and says that it is filled with quantum fields, propagating at every point in it, and that the energy of these fields cannot be zero. Is it possible that the curvature of spacetime, as in general relativity, results from the interaction of those quantum fields with the fields of quantum matter?
 A: I'm gonna write a pretty quick and not much detailed answer, but to give you a quick idea.
The first taught thing to quantize gravity is actually, that its quantum field behaves as a spin 2 particle with the same structure as the metric $g_{\mu\nu}$,  called the graviton .
So what people normally then do is say that the space-time curvature then is actually flat space-time + fluctuations of this quantum field, such that:
\begin{equation}
 g_{\mu\nu} = \eta_{\mu\nu} + \langle G_{\mu\nu}\rangle
\end{equation}
where we take the expected value of this fluctuation $G_{\mu\nu}$, the graviton.
A: 
Is it possible that the curvature of spacetime, as in general relativity, results from the interaction of those quantum fields with the fields of quantum matter?

I find this phrasing a little weird, as the fields of quantum matter are quantum fields, so I'll attempt to phrase it in a slightly different way to be more precise. I understand it in two possible ways, so I'll address both.
Semiclassical Gravity

Does the curvature of spacetime results from the energy-momentum of quantum fields?

That is what is usually understood in the framework known as semiclassical gravity, which attempts to understand some aspects of the interface of gravitation and quantum theory without fully quantizing gravity (usually, due to quantizing gravity being a fairly difficult task, although I've heard of some researchers who do side with the view that gravity is actually classical).
More specifically, in this framework it is common, for example, to consider the semiclassical Einstein equations, given in $\hbar = c = G = 1$ units by
$$G_{ab} = 8\pi \langle \hat{T}_{ab} \rangle_{\omega}, \tag{1}$$
where $\langle \hat{T}_{ab} \rangle_{\omega}$ is the expectation value of the renormalized stress-energy tensor in the state $\omega$. Notice this is a sort of "natural" generalization of the Einstein equations to the case in which you consider quantum matter, but still within the limit in which one assumes the quantum fluctuations on $\langle \hat{T}_{ab} \rangle_{\omega}$ to not be so large. Notice, for example, if one had picked $\omega$ to be a state in which the Sun is in a superposition of positions, then it is hard to believe Eq. (1) will be an appropriate description, since it tells one the gravitational field is that of the Sun being in the "midway" between both possible positions.
I should also mention that this framework is one extra step after doing Quantum Field Theory in Curved Spacetime, which corresponds to quantize fields on a previously chosen classical curved spacetime. In this situation, one assumes the quantum fields won't impact the curvature too harshly, so that their effects can be neglected. While this formalism does have some weird features for those used to QFT in flat spacetime (for example, one learns that not always it is possible to obtain a particle interpretation and that particles are actually observer-dependent), there is no difficulty in formulating the theory mathematically (at least in the free case, I'm not very acquainted with interacting QFTs in curved spacetime, although I think it has already been worked out with the aid of operator product expansions). This framework was used, for example, to understand the Unruh effect and the Hawking effect.
More information on quantum fields in curved spacetime can be found, e.g., in Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. The semiclassical Einstein equation is briefly discussed on the end of Sec. 4.6.
Quantum Gravity

Is the curvature of spacetime a manifestation of a quantum field?

This would mean the gravitational field to be a quantum field. As far as I know, most researchers in gravitation believe this, but often work with semiclassical gravity or QFT in curved spacetimes as useful approximations to an underlying, unknown theory.
To go a step further and actually quantize gravity is a different thing. Also, I'm less familiar with it, so this section might be a bit more sloppy. One can quantize GR and it has been done (John Donoghue has a number of works on this, such as these 2012 lecture notes), but quantum General Relativity is perturbatively non-renormalizable, which means usual QFT methods will lead to an inconsistent theory at high energies. One way of dealing with this without leaving the framework of QFT is: maybe the problem is with perturbation theory rather than gravity. In quite general terms, this is the concept of asymptotically safe quantum gravity, in which one still tries to treat gravity as a QFT, with the key point being the assumption GR should have a non-trivial UV fixed point in its renormalization flow (this can render a perturbatively non-renormalizable theory predictable even in the UV).
Due to gravity being background-independent, what one ideally does is to split the full metric as, e.g., $\hat{g}_{ab} = \bar{g}_{ab} + \hat{h}_{ab}$, where $\bar{g}_{ab}$ is a background classical metric (never specified completely) and $\hat{h}_{ab}$ represents the quantum fluctuations. In this way, one can treat $\hat{h}_{ab}$ as a quantum field in the background spacetime given by $\bar{g}_{ab}$. To have background independence, at the end of the day the results should not depend on $\bar{g}_{ab}$, which was then only used for technical reasons.
I'm definitely being quite rough in here, due to restrictions of memory, space, and knowledge. For more (and better) information, one can check, e.g., the recent books by Percacci, Reuter & Saueressig, the 2019 review article by Eichhorn (I think the article is open access, but in any case it is arXiv: 1810.07615 [hep-th]), among other references.
