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In this paper, from the $q$-gauge covariant condition we define the$q$-deformed Killing form and the second $q$-deformed Chern class for thequantum 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 thegeneralized $q$-deformed Killing forms are derived to prove the cocyclehierarchy formulas directly. At last, we construct the $q$-gauge covariantLagrangian and derive the $q$-deformed Yang-Mills equation. We find that thecomponents of the singlet and the adjoint representation are separated in the$q$-deformed Chern class, $q$-deformed cocycle hierarchy and the $q$-deformedLagrangian, although they are mixed in the commutative relations of BRSTalgebra

q-deformed Chern class, Chern-Simons and cocycle hierarchy