We construct the $q$-deformed Clifford algebra of $\mathfrak{sl}_2$ and study
its properties. This allows us to define the $q$-deformed noncommutative Weil
algebra for $U_q(\mathfrak{sl}_2)$ and the corresponding cubic Dirac operator.
In the classical case it was done by Alekseev and Meinrenken. We compute the
spectrum of the cubic element on finite-dimensional and Verma modules of
$U_q(\mathfrak{sl}_2)$ and the corresponding Dirac cohomology.
