For an endomorphism ${alpha}$ of R, in [1], a module $M_R$ is called ${alpha}$-compatible if, for any $m{in}M$ and $a{in}R$, ma = 0 iff $m{alpha}(a)$ = 0, which are a generalization of ${alpha}$-reduced modules. We study on the relationship between the quasi-Baerness and p.q.-Baer property of a module MR and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [2] and some results in [9]. In particular, we show: for an ${alpha}$-compatible module $M_R$ (1) $M_R$ is p.q.-Baer module iff $M[x;{alpha}]_{R[x;{alpha}]}$ is p.q.-Baer module. (2) for an automorphism ${alpha}$ of R, $M_R$ is p.q.-Baer module iff $M[x,x^{-1};{alpha}]_{R[x,x^{-1};{alpha}]}$ is p.q.-Baer module.