Could gravitons be quasiparticles like phonons?

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.