  In this paper, we study a family of generalized Weyl algebras $\{\A\}$ and
their polynomial extensions. We will show that the algebra $\A$ has a simple
localization $\A_{\mathbb{S}}$ when none of $p$ and $q$ is a root of unity. As
an application, we determine all the height-one prime ideals and the center for
$\A$, and prove that $\A$ is cancellative. Then we will determine the
automorphism group and solve the isomorphism problem for the generalized Weyl
algebras $\A$ and their polynomial extensions in the case where none of $p$ and
$q$ is a root of unity. We will establish a quantum analogue of the Dixmier
conjecture and compute the automorphism group for the simple localization
$(\mathcal{A}_{p}(1, 1, \K_{q}[s, t]))_{\mathbb{S}}$. Moreover, we will
completely determine the automorphism group for the algebra $\mathcal{A}_{p}(1,
1, \K_{q}[s, t])$ and its polynomial extension when $p\neq 1$ and $q\neq 1$.
