In our present investigation, we first introduce several new subclasses of analytic and bi-univalent functions by using a certain $q$-integral operator in the open unit disk $$\mathbb{U}=\{z: z\in \mathbb{C} \quad \text{and} \quad \left \vert z\right \vert <1\}.$$ By applying the Faber polynomial expansion method as well as the $q$-analysis, we then determine bounds for the $n$th coefficient in the Taylor-Maclaurin series expansion for functions in each of these newly-defined analytic and bi-univalent function classes subject to a gap series condition. We also highlight some known consequences of our main results.