A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring, G be an ordered monoid acting on R by $eta$ and R be G-compatible. It is shown that R is (left principally) quasi-Baer if and only if skew monoid ring $R_{eta}[G]$ is (left principally) quasi-Baer. If G is an abelian monoid, then R is (left principally) quasi-Baer if and only if the Cohn-Jordan extension $A(R,;eta)$ is (left principally) quasi-Baer if and only if left Ore quotient ring $G^{-1}R_{eta}[G]$ is (left principally) quasi-Baer.