Here is answer (YES) from Alexey Bolsinov who is one the main experts in these questions.
"The answer is YES
There is a very general construction allowing to construct an integrable system on more or less any coadjoint orbit for an arbitrary Lie algebra (non necessarily semisimple). This is a recent paper by Vinberg and Yakimova available in arxiv
http://arxiv.org/abs/math/0511498 Complete families of commuting functions for coisotropic Hamiltonian actions
In the particular case you are talking about (SEMI-SIMPLE g) the positive answer follows from 2 results:
1) the so-called shifts of polynomial invariants give a completely integrable system on a singular adjoint orbit O(b) in a semi simple Lie algebra G if and only if
the index of the centralizer of b coincides with the index of G
(my paper in Izvestija AN SSSR, 1991 and Acta Appl. Math. 1991), both available on my home page
Bolsinov A.V. Commutative families of functions related to consistent Poisson brackets// Acta Appl. Math., 24(1991), pp. 253-274.
I also conjectured that
this condition ind Cent (b) = ind G, in fact, holds true for all singular elements b\in G and checked it for G=sl(n) (in particular for all nilpotent)
2) This conjecture (widely known as Elashvili conjecture) has been proved for an arbitrary semi simple Lie algebra and for all elements (in fact the proof is easily reduced to nilpotent elements)
First, Elashvili did it by, in some sense, straightforward computation which in the most difficult case of e_8 involved some computer program (unpublished)
Recently a conceptual proof has been done by Jean-Yves Charbonnel and Moro (IMJ), Anne Moreau (available in arxiv)
http://arxiv.org/abs/1005.0831 The index of centralizers of elements of reductive Lie algebras
To the best of my knowledge, this is the only known universal way to construct an integrable system on an arbitrary orbit.
Remark: I am talking about classical integrable systems, not quantum. These systems can be quantized too, but this is another story. "