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 Zuminos 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.
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