  Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate
ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element
$w$ and they proved cluster monomials are contained in Lusztig's dual
semicanonical basis $S^{*}$. We give a set up for the quantization of their
results and propose a conjecture which relates the quantum cluster algebras to
the dual canonical basis ${B}^{up}$. In particular, we prove that the quantum
analogue $O_{q}[N(w)]$ of ${C}[N(w)]$ has the induced basis from ${B}^{up}$,
which contains quantum flag minors and satisfies a factorization property with
respect to the `$q$-center' of $O_{q}[N(w)]$. This generalizes Caldero's
results from ADE cases to an arbitary symmetrizable Kac-Moody Lie algebra.
