In this paper, we introduce the concept of $S^{2}$-weighted pseudo almost automorphy for stochastic processes. We study the existence and uniqueness of square-mean weighted pseudo almost automorphic solutions for the semilinear stochastic integral equation $x(t)=\int_{-\infty}^{t}a(t-s)[Ax(s)+f(s,x(s))]ds+\int_{-\infty}^{t}a(t-s)\varphi(s,x(s))dw(s), \ t\in\mathbb{R}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family on a Hilbert space $H$, $w(t)$ is the two-sided $Q$-Wiener process, $f,\varphi: \mathbb{R}\times L^{2}(P,H)\rightarrow L^{2}(P,H)$ are two $S^{2}$-weighted pseudo almost automorphic functions.