  For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, we construct a
family of weighted quivers $Q_m(\mathfrak{g})$ ($m \geq 2$) whose cluster
modular group $\Gamma_{Q_m(\mathfrak{g})}$ contains the Weyl group
$W(\mathfrak{g})$ as a subgroup. We compute explicit formulae for the
corresponding cluster $\mathcal{A}$- and $\mathcal{X}$-transformations. As a
result, we obtain green sequences and the cluster Donaldson-Thomas
transformation for $Q_m(\mathfrak{g})$ in a systematic way when $\mathfrak{g}$
is of finite type. Moreover if $\mathfrak{g}$ is of classical finite type with
the Coxeter number $h$, the quiver $Q_{kh}(\mathfrak{g})$ ($k \geq 1$) is
mutation-equivalent to a quiver encoding the cluster structure of the higher
Teichm\"uller space of a once-punctured disk with $2k$ marked points on the
boundary, up to frozen vertices. This correspondence induces the action of
direct products of Weyl groups on the higher Teichm\"uller space of a general
marked surface. We finally prove that this action coincides with the one
constructed in [GS18] from the geometrical viewpoint.
