  We present a solution of the $(A_2+A_1)^{(1)}$ $q$-Painlev\'{e} equation in
terms of the $\mu$-function. The $\mu$-function introduced by Zwegers is the
most fundamental object in the study of mock theta functions. The results of
this paper give us an expectation that the theory of mock modular forms and the
$\tau$-functions of discrete integrable systems are closely related.
