Coherently driven phonons are conventionally described classically. Nonlinear phonon-phonon interactions, however, allow an infrared drive to reshape the fluctuation spectrum and generate substantial nonthermal enhancement of phonon fluctuations, motivating a description beyond the coherent displacement alone. We benchmark classical and semiclassical approximations against the full quantum theory and find that the semiclassical treatment substantially improves agreement with the quantum dynamics. We propose nonequilibrium spontaneous Raman scattering as a sensitive probe of these effects and apply our framework to recent experiments suggesting quantum nonlinear phononics involving the E₁ᵤ and E_2g modes in multilayer graphene. At very strong pump fields, driving the E₁ᵤ mode significantly modifies the E_2g Raman spectrum; however, reproducing the reported response at experimental field strengths requires accounting for pump-induced Raman matrix element dynamics or alternative energy pathways into the E_2g mode. Our findings extend nonlinear phononics beyond average atomic displacements to the dynamics of lattice fluctuations, highlighting the interplay between coherent motion and quantum correlations in driven materials.