  For any complex reflection group $G=G(m,p,n)$, we prove that the
$G$-invariants of the division ring of fractions of the $n$:th tensor power of
the quantum plane is a quantum Weyl field and give explicit parameters for this
quantum Weyl field. This shows that the $q$-Difference Noether Problem has a
positive solution for such groups, generalizing previous work by Futorny and
the author. Moreover, the new result is simultaneously a $q$-deformation of the
classical commutative case, and of the Weyl algebra case recently obtained by
Eshmatov et al.
  Secondly, we introduce a new family of algebras called quantum OGZ algebras.
They are natural quantizations of the OGZ algebras introduced by Mazorchuk
originating in the classical Gelfand-Tsetlin formulas. Special cases of quantum
OGZ algebras include the quantized enveloping algebra of $\mathfrak{gl}_n$ and
quantized Heisenberg algebras. We show that any quantum OGZ algebra can be
naturally realized as a Galois ring in the sense of Futorny-Ovsienko, with
symmetry group being a direct product of complex reflection groups
$G(m,p,r_k)$.
  Finally, using these results we prove that the quantum OGZ algebras satisfy
the quantum Gelfand-Kirillov conjecture by explicitly computing their division
ring of fractions.
