In this article, we consider new subclasses of analytic and bi-univalent functions associated with the q-Srivastava-Attiya operator in the open unit disk. We obtain coefficient bounds for the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ of the functions of these new subclasses. Furthermore, we establish the Fekete-Szegö inequality for functions in the classes ${\mathcal{T}}_{\tau ,\mathfrak{q},\alpha }^{ϵ}(\psi ),{\mathcal{K}\mathcal{H}}_{\tau ,\mathfrak{q},\alpha }^{ϵ}(\delta ,\psi ),\text{and}{\mathcal{A}}_{\tau ,\mathfrak{q},\alpha }^{ϵ}(\delta ,\psi )$.