In this paper, we first solve the Dirichlet problem for $\alpha$-Hermite-Yang-Mills-Higgs equations on $I_{\pm}$-Higgs bundles over a compact generalized K\"{a}hler manifold. Then we prove that the $\alpha$-semi-stability on $I_{\pm}$-Higgs bundles over closed generalized K\"{a}hler manifolds implies the existence of approximate $\alpha$-Hermite-Yang-Mills-Higgs structure.