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 Bäcklund transformations for Painlevé equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra for any symmetrizable generalized Cartan matrix. 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 Bäcklund transformations of $q$-Painlevé 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.