In this paper, we prove the existence of the approximate $(\sigma ,\tau )$-Hermitian Yang-Mills structure on the $(\sigma ,\tau )$-semi-stable quiver bundle $\mathcal{R}=(\mathcal{E},\varphi )$ over compact Gauduchon manifolds. An interesting aspect of this work is that the argument on the weakly $L_1^2$-subbundles is different from [Álvarez-Cónsul and García-Prada, Comm. Math. Phys., 2003] and [Hu-Huang, J. Geom. Anal., 2020].