Cu-63,Cu-65 NQR-NMR relaxation measurements in CeCu6-xAux for x=0, x=0.1, and x=0.8 are used to derive insights on temperature (T) and magnetic-field (H) dependences of the spin dynamics around the quantum critical point (QCP). The relaxation rate 2W is related to the k integrated, low energy, generalized susceptibility chi(k,omega,H,T-->0). For x=0 a Fermi-liquid behavior is detected, while for x=0.8 the temperature dependence of 2W is the one expected for a nearly antiferromagnetic metal ordering at T-N=2.2 K. Instead, for x=0.1, around the QCP, a response function of the form suggested by neutron scattering, namely of two-dimensional character with anomalous exponent and (omega/T) scaling, is found to explain the main experimental findings. An effect is observed in the low-temperature range for Hgreater than or similar to1 T, with a crossover to a gapped phase for the spin excitations at a field-dependent temperature.