In this study, using the ideas of subordination and the quantum-difference operator, we established a new subclass ${\mathcal{S}}^{\ast }\left(\delta ,\sigma ,q\right)$ of $q$-starlike functions and the subclass ${\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }\left(\delta ,\sigma ,q\right)$ of $q$-starlike bi-univalent functions associated with the vertical strip domain. We examined sharp bounds for the first two Taylor-Maclaurin coefficients, sharp Fekete-Szegö type problems, and coefficient inequalities for the function $h$ that belong to ${\mathcal{S}}^{\ast }\left(\delta ,\sigma ,q\right)$, as well as sharp bounds for the inverse function $h$ that belong to ${\mathcal{S}}^{\ast }\left(\delta ,\sigma ,q\right)$. We also investigated some results for the class of bi-univalent functions ${\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }\left(\delta ,\sigma ,q\right)$ and well-known corollaries were also highlighted to show connections between previous results and the findings of this paper.