  We determine the explicit quantum ordering for a special class of quantum
geodesic functions corresponding to geodesics joining exactly two orbifold
points or holes on a non-compact Riemann surface. We discuss some special cases
in which these quantum geodesic functions form sub--algebras of some abstract
algebras defined by the reflection equation and we extend our results to the
quantisation of matrix elements of the Fuchsian group associated to the Riemann
surface in Poincar\'e uniformization. In particular we explore an interesting
relation between the deformed $U_q(\mathfrak{sl}_2)$ and the Zhedanov algebra
AW(3).
