Y. Hirano introduced the concept of a quasi-Armendariz ring which extends both Armendariz rings and semiprime rings. A ring R is called quasi-Armendariz if $a_iRb_j$ = 0 for each i, j whenever polynomials $f(x);=;sum_{i=0}^ma_ix^i$, $g(x);=;sum_{j=0}^mb_jx^j;{in};R[x]$ satisfy f(x)R[x]g(x) = 0. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if R is a semiprime ring with an epimorphism $sigma$, then f(x)R[x; $sigma$]g(x) = 0 implies $a_iR{sigma}^{i+k}(b_j)=0$ for any integer k $geq$ 0 and i, j, where $f(x);=;sum_{i=0}^ma_ix^i$, $g(x);=;sum_{j=0}^mb_jx^j;{in};R[x,;{sigma}]$. Moreover, we extend this property to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define $sigma$-skew quasi-Armendariz rings for an endomorphism $sigma$ of a ring R. Then we study several extensions of $sigma$-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and $sigma$-skew Armendariz rings.