A ring with identity is generalized quasi-Baer if for any ideal $I$ of $R$, the right annihilator of $I^n$ is generated by an idempotent for some positive integer $n$, depending on $I$. We study the generalized quasi-Baerness of $R[x;\sigma;\delta]$ over a generalized quasi-Baer ring $R$ where $\sigma$ is an automorphism of $R$.