Recently, biomolecules which are labeled by azide or thiocyanate groups in solutions and proteins have been studied to examine microscopic environment around a solute by nonlinear infrared (IR) spectroscopy. In this study, we have performed two-dimensional infrared (2D-IR) spectroscopy to investigate the vibrational frequency fluctuations of two different azide-derivatized amino acids, Ala (N3-Ala) and Pro (N3-Pro), and N3(-) in water. From the 2D-IR experiments, it was found that the frequency-frequency time correlation function (FFTCF) of solute can be modeled by a delta function plus an exponential function and constant. FFTCF for each probe molecule has a decay component of about 1 ps, and this result suggests that the stretching mode of the covalently bonded azide group is sensitive to the fluctuations of hydrogen bond network system, as found in previous studies of N3(-) in water. In contrast to FFTCF of N3(-), FFTCF of the azide-derivatized amino acids contains static component. This static component may reflect dynamics of water affected by the solutes or the structural fluctuations of the solute itself. We also performed the IR pump-probe measurements for the probe molecules in water in order to investigate vibrational energy relaxation (VER) and reorientational relaxation. It was revealed that the charge fluctuations in the azide group are significant for the VER of this mode in water, reflecting that the VER rate of N3(-) is faster than those of the azide-derivatized amino acids. While the behaviors of the anisotropy decay of N3-Ala and N3(-) are similar to each other, the anisotropy decay of N3-Pro contains much slower decaying component. By considering the structural difference around the vibrational probe between N3-Ala and N3-Pro, it is suggested that the structural freedom of the probe molecules can affect the reorientational processes.