[Edit: removed an attribution after Shizhuo's correction]
Kremnizer gave a nice course where he worked through the examples of G=SL_2, G/B=CP^1 in complete detail. I have some incomplete notes if you want to email me (I'd rather not post them online since they are still being revised).
To answer your question briefly about what the notion of quantum differential operators are for the flag variety, here is a rough outline:
1) First, define quantum differential operators on G. This is done by constructing the so-called Heisenberg double (just another name in this specific situation for the semi-direct product, also called smash prodcut) D(U_q,O_q) of the quantum group U_q with its dual Hopf algebra O_q, where U_q acts on O_q by the left-regular action (X.f)(y):=f(S(X)y), here X is the antipode.
2) O_q(G) has a sub-algebra called O_q(B) which is a quantization of the functions on the Borel.
3) In sufficiently nice cases in algebraic geometry, one can identify D(X/G)-modules where X is some variety with G-action as D(X)-modules M, together with an O(G) co-action, and a compatibility condition. The case G/B is such a situation classically, so one defines D_q(G/B)-modules to be D_q(G)-modules with a O_q(B) co-action plus compatibility.
By the way, there are some papers of Varagnolo and Vasserot, notably http://arxiv.org/abs/math/0603744, which discuss D_q(G) and might introduce you to some tricks people use in the area.
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