Incoherent quasi-elastic neutron scattering (QENS) has been used to measure the dynamics of water molecules in solutions of a model protein backbone, N-acetyl-glycine-methylamide (NAGMA), as a function of concentration, for comparison with results for water dynamics in aqueous solutions of the N-acetyl-leucine-methylamide (NALMA) hydrophobic peptide at comparable concentrations. From the analysis of the elastic incoherent structure factor, we find significant fractions of elastic intensity at high and low concentrations for both solutes, which corresponds to a greater population of protons with rotational time scales outside the experimental resolution (> 13 ps). The higher-concentration solutions show a component of the elastic fraction that we propose is due to water motions that are strongly coupled to the solute motions, while for low-concentration solutions an additional component is activated due to dynamic coupling between inner and outer hydration layers. An important difference between the solute types at the highest concentration studied is found from stretched exponential fits to their experimental intermediate scattering functions, showing more pronounced anomalous diffusion signatures for NALMA, including a smaller stretched exponent beta and a longer structural relaxation time tau than those found for NAGMA. The more normal water diffusion exhibited near the hydrophilic NAGMA provides experimental support for an explanation of the origin of the anomalous diffusion behavior of NALMA as arising from frustrated interactions between water molecules when a chemical interface is formed upon addition of a hydrophobic side chain, inducing spatial heterogeneity in the hydration dynamics in the two types of regions of the NALMA peptide. We place our QENS measurements on model biological solutes in the context of other spectroscopic techniques and provide both confirming as well as complementary dynamic information that attempts to give a unifying molecular view of hydration dynamics signatures near peptides and proteins.