  We compute the Bonahon-Wong-Yang quantum invariant for self-diffeomorphisms
of the four-puncture sphere explicitly, based on the representation theory of
the Checkov-Fock algebra. As an application of the computation, we verify the
volume conjecture proposed by Bonahon-Wong-Yang for the four puncture sphere
bundles with some technical conditions. Together with the one-puncture torus
bundles, this is the first family of examples for which the conjecture is
verified providing evidence for the conjecture
