  By using the cluster perturbation theory, we investigate the effects of the
local electron-phonon interaction in the quantum spin Hall topological
insulator described by the half-filled Kane-Mele model on an honeycomb lattice.
Starting from the topological non trivial phase, where the minimal gap is
located at the two inequivalent Dirac points of the Graphene, $\text{K}$ and
$\text{K}'$, we show that the coupling with quantum phonons induces a
topological-trivial quantum phase transition through a gap closing and
reopening in the $\text{M}$ point of the Brillouin zone. The average number of
fermions in this point turns out to be a direct indicator of the quantum
transition pointing out a strong hybridization between the two bare
quasiparticle bands of the Kane-Mele model. By increasing the strength of
charge-lattice coupling, the phonon Green's propagator displays a two peak
structure: the one located at the lowest energy exhibits a softening that is
maximum around the topological transition. Numerical simulations provide also
evidence of several kinks in the quasiparticle dispersion caused by the
coupling of the electrons with the bosonic lattice mode.
