Could gravitons be quasiparticles like phonons? The Einstein field equation
$$
G_{\mu \nu} +  \Lambda g_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu}
$$
basically says: 
$$
\text{curvature of spacetime} \sim \text{stress-energy tensor}
$$
The stress-energy tensor is most often written in the perfect compressible fluid form, so we can say 
$$
\text{curvature of spacetime} \sim \text{perfect fluid motion}
$$
If we linearize both sides of this "equation" we get
$$
\text{gravitational waves} \sim \text{sound waves}
$$
And if we quantize both sides, we get
$$
\text{gravitons} \sim \text{phonons}
$$
Could therefore gravitons be quasiparticles like phonons? 
 A: This idea has been seriously considered by many physicists in the past.  What you are proposing sounds very similar to Sakharov's idea of induced gravity.
More recent theories like loop quantum gravity and the related theory of causal dynamical triangulation have also proposed that spacetime has a discrete, lattice-like microscopic structure at the Planck scale, which coarse-grains to the smooth pseudo-Riemannian manifold of general relativity at much larger scales:

The main output of the theory is a physical picture of space where space is granular. ... More precisely, space can be viewed as an extremely fine fabric or network "woven" of finite loops ... called spin networks. ... The predicted size of this structure is the Planck length, which is approximately $10^{−35}$ meters. ... Therefore, LQG predicts that not just matter, but space itself, has an atomic structure.

Another theory, called entropic gravity,

implies that gravity is not a fundamental interaction, but an emergent phenomenon which arises from the statistical behavior of microscopic degrees of freedom encoded on a holographic screen.

In either such picture, gravitons would presumably be quasiparticle-like "macroscopic" (relative to the Planck scale) collective excitations of the discrete Planck-scale degrees of freedom, just like phonons are.
A: 
The Einstein field equation $G_{\mu \nu} +  \Lambda g_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu}$ basically says: curvature of spacetime ~ stress-energy tensor.

IMHO that's too simplistic. See this answer for what I think it says. 

The stress-energy tensor is most often written in the perfect compressible fluid form, so we can say curvature of spacetime ~ perfect fluid motion.

No. It isn't a fluid. It's more like a gin-clear ghostly elastic solid. Hence there's a shear-stress term in the stress-energy-momentum tensor which "describes the density and flux of energy and momentum in spacetime".

Public domain image by Maschen, based on an image by Bamse

If we linearize both sides of this "equation" we get gravitational waves ~ sound waves. 

Gravitational waves are sometimes described as being like sound waves. Note though that sound waves in the air are longitudinal waves, whilst gravitational waves are quadrupole waves. Also note that a gravitational wave is a graviton. That's something different to the messenger-particle graviton people talk about in the context of quantum gravity. That's akin to the messenger-particle photon, which is a virtual photon. It's not a real photon. It's an abstract thing that only exists in the mathematics of the model.     

And if we quantize both sides, we get gravitons ~ phonons. Could therefore gravitons be quasiparticles like phonons? 

Yes and no. A phonon is a quasiparticle. Quasiparticles are "emergent phenomena that occur when a microscopically complicated system such as a solid". When you look into a photon you find things like this: “We can scarcely avoid the inference that light consists in the traverse undulations of the same medium which is the cause of electric and magnetic phenomena”. The transverse-wave aspect suggests a solid, which suggests that the photon is not totally unlike a phonon. Particularly since there are acoustic and optical phonons. A gravitational wave is a graviton, and whilst it isn't the same thing as a photon or a phonon, it's not totally different. A phonon is a wave in a crystal lattice, a gravitational wave is a wave in space. So is a photon. 
