  In this paper, we establish the Hitchin--Kobayashi correspondence for the
$I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,\phi)$ over a compact
generalized K\"{a}hler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon
with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is
$(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an
$(\alpha,\sigma,\tau)$-Hermitian--Einstein metric.
