  We construct a new solution to the tetrahedron equation and the
three-dimensional (3D) reflection equation by extending the quantum cluster
algebra approach by Sun and Yagi concerning the former. We consider the
Fock-Goncharov quivers associated with the longest elements of the Weyl groups
of type $A$ and $C$, and investigate the cluster transformations corresponding
to changing a reduced expression into a `most distant' one. By devising a new
realization of the quantum $y$-variables in terms of $q$-Weyl algebra, the
solutions are extracted as the operators whose adjoint actions yield the
cluster transformations of the quantum $y$-variables. Explicit formulas of
their matrix elements are also derived for some typical representations.
