Quasi-ice-like hydration waters on the ice-binding surface (IBS) of an antifreeze protein (AFP) commonly exhibit sluggish dynamics especially at low temperatures. In this work, we have analyzed molecular dynamics (MD) simulation trajectories at two different temperatures for Tenebrio molitor antifreeze protein (TmAFP) to explore whether the unique quasi-ice-like structuring of hydration water has any impact on making their dynamics slower on the IBS of the protein. Our calculation reveals that, as translational dynamics is coupled with the conformational fluctuations, hydration water on the IBS exhibits sluggish translational motion due to reduced flexibility of the IBS compared to that on the non-ice-binding surface (NIBS) of the protein. Interestingly, it is noticed that rotational motion of hydration water is not coupled with the conformational fluctuations of the surfaces. In that case, structural relaxations of the protein-water (PW) and water-water (WW) hydrogen bonds compete with each other to make the rotational dynamics of hydration water around the IBS either faster or slower with respect to those around the NIBS. At low temperature, the slower structural relaxation of water-water hydrogen bonds dominates and imparts sluggish rotational motion of the hydration water on the IBS of the protein. The slower structural relaxation of water-water hydrogen bonds and hence the retarded rotational dynamics, despite the weak short-lived PW hydrogen bonds on the IBS, is clearly a manifestation of the rigid quasi-ice-like structure of the hydration shell on that surface.