Using Chebyshev polynomials and q-differential operator, we introduce a novel class of bi-univalent functions within the open unit disk. Initial coefficient bounds for this class are established, emphasizing their significance in complex analysis. Analyticity ensures the representation of these functions through convergent power series, offering a robust tool for comprehending their behavior. The condition of univalence guarantees the one-to-one nature of these functions, preventing multiple mappings of the same point. Researchers employ bi-univalent functions to delve into diverse aspects of complex analysis, unraveling their properties and applications across various mathematical contexts. The exploration encompasses the investigation of geometric properties, including Fekete-Szegö inequality and coefficient bounds, unraveling the intricate interplay between analytic and geometric characteristics. In summary, the study of bi-univalent functions contributes to the depth of complex analysis, providing a nuanced understanding of the relationships between analyticity and univalence. This exploration lays the foundation for advancements in mathematical theory and applications.