There exist a large family of noncommutative spaces that arise from the quantum matrices. These algebraic objects $q$-deform the coordinate rings of certain varieties. For example, take quantum $SU(2)$, this is the algebra $< a,b,c,d >$ quotiented by the ideal generated by
$$
ab−qba, ~~ ac−qca, ~~ bc−cb, ~~ bd−qdb, ~~ cd−qdc, ~~ ad−da−(q−q^{−1})bc, 
$$
and the "q-det" relation
$$
ad−qbc−1
$$
where $q$ is some complex number. Clearly, when $q=1$ we get back the coordinate ring of $SU(2)$. In the classical case $S^2 = SU(2)/U(1)$ (the famous Hopf fibration). This generalises to the q-case: the $U(1)$-action generalises to a $U(1)$-coaction with an invariant subalgebra that q-deforms the coordinate algebra of $S^2$ - the famous Podles sphere. There exist such q-matrix deformations of all flag manifolds. 


Since all such manifolds are Kahler, we can also apply Kontsevich deformation to them to obtain a q-defomation. My question is: What is the relationship between these two approaches?


Alternatively, we can apply Kostant-Souriau geometric quantization to a flag manifold. How does alegbra relate to its q-matrix deformation? 



Kontsevich, and Geometric, Quantization and the Podles sphere
As far as I understand, the flag manifolds with Kahler structures mentioned in the question are simply coadjoint orbits of compact Lie groups with the Kirillov-Kostant-Souriau bracket, so their quantizations will yield quotients of the usual enveloping algebra $U(g)$ and will not have to do with quantum groups. I suppose that the q-spaces discussed in the question are meant to be q-deformations of these. 


Here are some papers about this: arXiv:math/0206049 and arXiv:math/9807159. This is a rather subtle business: e.g., it is explained that the 2-parameter deformations (similar to the 2-parameter family of Podles spheres) do not always exist, although they do exist in type A
and in many other cases, e.g. if orbits are symmetric spaces.


