Proton nuclear magnetic resonance (1H NMR) experiments have been performed to measure the spin-lattice, T1, and spin-spin, T2, relaxation times of the three functional groups in water/methanol mixtures at different methanol molar fractions (XMeOH=0, 0.04, 0.1, 0.24, 0.5, 1) as a function of temperature in the range 205 K<T<295 K. The measured relaxation times in the mixtures, at all the methanol molar fractions, are faster than those of pure water and methanol because of strong interactions, resulting in a complex hydrogen bonding dynamics that determines their thermodynamic properties. In particular, we observe how the interplay between hydrophobicity and hydrophilicity changes with temperature and influences the peculiar thermal behavior of the NMR relaxation times of the solution. The obtained results are interpreted in terms of the existence of stable water-methanol clusters at high temperature whereas, upon cooling to low temperature, clusters of single species are present in the mixture.