We report the observation of the quantum walks of a phonon, a vibrational quantum, in a trapped-ion crystal. By employing the capability to prepare and observe the localized wave packet of a phonon, the propagation of a single radial local phonon in a four-ion linear crystal is observed with single-site resolution. The results show an agreement with numerical calculations, indicating the predictability and reproducibility of the phonon system. These characteristics may contribute advantageously in the advanced studies of quantum walks, as well as boson sampling and quantum simulation.