This is shown in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. The result you ask for is Proposition 63 in Chapter 11. I'd expand more upon this but I have to give a talk shortly, and anyway it's all there in the book.
Edit: I guess I should say that the point here is the representation theory. Classically, we can think of the algebra of functions on the sphere as the representation of $SU(N)$ induced by the trivial representation of the subgroup $U(N-1)$. Then Frobenius Reciprocity tells you which representations of $SU(N)$ occur as summands in the function algebra, and hence which matrix coefficients generate it. These things all transfer over to the quantum setting because they are phrased in terms of representations.