Shifted quantum affine algebras are quantum groups parameterized by a coweight of the underlying Lie algebra. In 2022, Hernandez introduced a category O of representations of these algebras, and in 2024, Geiss–Hernandez–Leclerc proved that the Grothendieck ring of this category O admits a cluster algebra structure. In this talk, after introducing the necessary background, I will explain how to construct a quantization of this cluster algebra, leading to a definition of the quantum Grothendieck ring for category O. If time permits, I will also discuss some applications and directions for future research.