For a closed surface of genus 1 and 2 the q-skein algebra (of type A1) admits a description in terms of 1 resp. 3 commuting difference operators acting on commutative polynomials in 1 resp. 3 variables. This endows it with a stucture of a completely integrable quantum mechanical system. We argue that in higher genus analogous description exists, involving 3g - 3 commuting operators acting on certain noncommutative polynomials. This noncommutativity first shows up is genus 3, where the polynomials in question form the Askey-Wilson algebra AW(3).