  We shall construct the quantized q-analogues of the birational Weyl group
actions arising from nilpotent Poisson algebras, which are conceptual
generalizations, proposed by Noumi and Yamada, of the B\"acklund
transformations for Painlev\'e equations. Consider a quotient Ore domain of the
lower nilpotent part of a quantized universal enveloping algebra of arbitrary
symmetrizable Kac-Moody type. Then non-integral powers of the image of the
Chevalley generators generate the quantized q-analogue of the birational Weyl
group action. Using the same method, we shall reconstruct the quantized
B\"acklund transformations of q-Painlev\'e equations constructed by Hasegawa.
We shall also prove that any subquotient integral domain of a quantized
universal enveloping algebra of finite or affine type is an Ore domain.
