The study of holomorphic functions has been recently extended through the application of diverse techniques, among which quantum calculus stands out due to its wide-ranging applications across various scientific disciplines. In this context, we introduce a novel q-differential operator defined via the generalized binomial series, which leads to the derivation of new classes of quantum-convex (q-convex) functions. Several specific instances within these classes were explored in detail. Consequently, the boundary values of the Hankel determinants associated with these functions were analyzed. All graphical representations and computational analyses were performed using Mathematica 12.0.•These classes are defined by utilizing a new q-differential operator.•The coefficient values | a i | ( i = 2 , 3 , 4 ) are investigated.•Toeplitz determinants, such as the second T 2 ( 2 ) and the third T 3 ( 1 ) order inequalities, are calculated.