Summary: We introduce the concept of quantum integration for interval-valued functions, and establish new \(q\)-Hermite-Hadamard and \(q\)-Hermite-Hadamard-Fejer inequalities for left and right \(h\)-convex interval-valued functions. The findings generalize known results in the literature and serve as a foundation for future studies in inequalities for interval-valued functions and interval differential equations. The results are illustrated with examples.