  A Riemann-Hilbert problem for a $q$-difference Painlev\'e equation, known as
$q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective
correspondence between the transcendental solutions of
$q\textrm{P}_{\textrm{IV}}$ and corresponding data on an associated
$q$-monodromy surface. We also construct the moduli space of
$q\textrm{P}_{\textrm{IV}}$ explicitly.
