We use gauge-gravity duality to study the temperature dependence of the zero sound mode and the fundamental matter diffusion mode in the strongly coupled $\mathcal{N}=4$ $SU({N}_{c})$ supersymmetric Yang-Mills theory with ${N}_{f}$ $\mathcal{N}=2$ hypermultiplets in the ${N}_{c}\ensuremath{\gg}1$, ${N}_{c}\ensuremath{\gg}{N}_{f}$ limit, which is holographically realized via the D3/D7 brane system. In the high density limit $\ensuremath{\mu}\ensuremath{\gg}T$, three regimes can be identified in the behavior of these modes, analogous to the collisionless quantum, collisionless thermal, and hydrodynamic regimes of a Landau Fermi liquid. The transitions between the three regimes are characterized by the parameters $T/\ensuremath{\mu}$ and $(T/\ensuremath{\mu}{)}^{2}$, respectively, and in each of these regimes the modes have a distinctively different temperature and momentum dependence. The collisionless-hydrodynamic transition occurs when the zero sound poles of the density-density correlator in the complex frequency plane collide on the imaginary axis to produce a hydrodynamic diffusion pole. We observe that the properties characteristic of a Landau Fermi-liquid zero sound mode are present in the D3/D7 system despite the atypical ${T}^{6}/{\ensuremath{\mu}}^{3}$ temperature scaling of the specific heat and an apparent lack of a directly identifiable Fermi surface.