In this investigation, the q-difference operator and the Sălăgean q-differential operator are utilized to establish novel subclasses of analytical bi-close-to-convex functions. We determine the general Taylor–Maclaurin coefficient of the functions in this class using the Faber polynomial method. We demonstrate the unpredictable behaviour of initial coefficients a2, a3 and investigate the Fekete–Szegő problem a3−a22 for the subclasses of bi-close-to-convex functions. To highlight the connections between existing knowledge and new research, certain known and unknown corollaries are also highlighted.