  Using free field representation of quantum affine algebra
$U_q(\widehat{sl_2})$, we investigate the structure of the Fock modules over
$U_q(\widehat{sl_2})$. The analisys is based on a $q$-analog of the BRST
formalism given by Bernard and Felder in the affine Kac-Moody algebra $\widehat
{sl_2}$. We give an explicit construction of the singular vectors using the
BRST charge. By the same cohomology analysis as the classical case ($q=1$), we
obtain the irreducible highest weight representation space as a nontrivial
cohomology group. This enables us to calculate a trace of the $q$-vertex
operators over this space.
