  We perform thermodynamic and inelastic neutron scattering (INS) measurements
to study the lattice dynamics (phonons) of a cubic collinear antiferromagnet
Cu$_3$TeO$_6$ which hosts topological spin excitations (magnons). While the
specific heat and thermal conductivity results show that the thermal transport
is dominated by phonons, the deviation of the thermal conductivity from a pure
phononic model indicates that there is a strong coupling between magnons and
phonons. In the INS measurements, we find a mode in the excitation spectra at
4.5 K, which exhibits a slight downward dispersion around the Brillouin zone
center. This mode disappears above the N\'{e}el temperature, and thus cannot be
a phonon. Furthermore, the dispersion is distinct from that of a magnon.
Instead, it can be explained by the magnon-polaron mode, which is new
collective excitations resulting from the hybridization between magnons and
phonons. We consider the suppression of the thermal conductivity and emergence
of the magnon-polaron mode to be evidence for magnon-phonon coupling in
Cu$_3$TeO$_6$.
