  We apply the differential Galois theory for difference equations developed by
Hardouin and Singer to compute the differential Galois group for a second-order
linear $q$-difference equation with rational function coefficients. This Galois
group encodes the possible polynomial differential relations among the
solutions of the equation. We apply our results to compute the differential
Galois groups of several concrete $q$-difference equations, including for the
colored Jones polynomial of a certain knot.
