  In this paper, from the $q$-gauge covariant condition we define the
$q$-deformed Killing form and the second $q$-deformed Chern class for the
quantum group $SU_{q}(2)$. Developing Zumino's method we introduce a
$q$-deformed homotopy operator to compute the $q$-deformed Chern-Simons and the
$q$-deformed cocycle hierarchy. Some recursive relations related to the
generalized $q$-deformed Killing forms are derived to prove the cocycle
hierarchy formulas directly. At last, we construct the $q$-gauge covariant
Lagrangian and derive the $q$-deformed Yang-Mills equation. We find that the
components of the singlet and the adjoint representation are separated in the
$q$-deformed Chern class, $q$-deformed cocycle hierarchy and the $q$-deformed
Lagrangian, although they are mixed in the commutative relations of BRST
algebra.
