Phonons (quanta of collective vibrations) are a major source of energy dissipation and drive some of the most relevant properties of materials. In nanotechnology, phonons severely affect light emission and charge transport of nanodevices. While the phonon response is conventionally considered an inherent property of a nanomaterial, here we show that the dipole-active phonon resonance of semiconducting (CdS) nanocrystals can be drastically reshaped inside a terahertz plasmonic nanocavity, via the phonon strong coupling with the cavity vacuum electric field. Such quantum zero-point field can indeed reach extreme values in a plasmonic nanocavity, thanks to a mode volume well below λ3/107. Through Raman measurements, we find that the nanocrystals within a nanocavity exhibit two new "hybridized" phonon peaks, whose spectral separation increases with the number of nanocrystals. Our findings open exciting perspectives for engineering the optical phonon response of functional nanomaterials and for implementing a novel platform for nanoscale quantum optomechanics.