  We study a family of integrals parameterised by $ N = 2,3,\dots $
generalising the Askey-Wilson integral $ N=2 $ which has arisen in the theory
of $q$-analogs of monodromy preserving deformations of linear differential
systems and in theory of the Baxter $Q$ operator for the $ XXZ $ open quantum
spin chain. These integrals are particular examples of moments defined by
weights generalising the Askey-Wilson weight and we show the integrals are
characterised by various $ (N-1) $-th order linear $q$-difference equations
which we construct. In addition we demonstrate that these integrals can be
evaluated as a finite sum of $ (N-1) $ $ BC_{1} $-type Jackson integrals or $
{}_{2N+2}\varphi_{2N+1} $ basic hypergeometric functions.
