We report Ge-73 nuclear magnetic resonance/nuclear quadrupole resonance (NMR/NQR) measurements on the itinerant ferromagnetic superconductor UGe2 at ambient pressure (P = 0) and P = 1.3 GPa. Measurements of the nuclear spin-lattice relaxation rate 1/T-1 of the Ge-73 NMR at P = 0 have revealed a T1T = constant behaviour well below T-Curie = 52 K, evidencing the presence of a non-zero density of states at the Fermi level for the up-and down-spin bands in the ferromagnetic state. At P = 1.3 GPa, where the ferromagnetic transition is reduced to T-Curie = 26 K, the dependence on temperature (T) of NQR 1/T-1 at zero field has demonstrated the onset of the superconducting transition at T, = 0.55 K. The lack of a coherence peak in (1/T-1) just below T,, followed by a T-3-like behaviour, provide compelling evidence for the unconventional nature of the superconducting state that coexists with the ferromagnetic state on a microscopic scale in UGe2.