For every 0 < q < 1 and 0 ≤ α < 1, we introduce a class of analytic functions f on the open unit disc D with the standard normalization f (0) = 0 = f' (0) − 1 and satisfying 1 1 − α z(D q f)(z) h(z) − α − 1 1 − q ≤ 1 1 − q , (z ∈ D), where h ∈ S * q. This class is denoted by K q (α), so called the class of q-close-to-convex-functions of order α. In this paper, we study some geometric properties of this class. In addition, we consider the famous Bieberbach conjecture problem on coefficients for the class K q (α). We also find some sufficient conditions for the function to be in K q (α) for some particular choices of the functions h. Finally, we provide some applications on q-analogue of Gaussian hypergeometric function.