We have performed field-swept Co-59 NMR measurements on the "exotic" Fin material, Ca3Co2O6, crystallizing in K4CdCl6-type rhombohedral structure, believed to exhibit two magnetic transition temperatures (around 24 and 12 K). This is the first NMR investigation of this family of compounds. We are able to detect the NMR signal below about 15 K, which is conclusive proof for the existence of nonmagnetic Co ions (Co3+-low-spin 3d (6), in addition to magnetic Co ions) at least in a narrow temperature range around 10 K, thereby resolving a controversy on this issue in the literature. In addition, we find that the spin-lattice and spin-spin relaxation times undergo a dramatic increase below about 12 K, with a profound influence on the intensity of the NMR signal as a function of temperature, providing microscopic evidence for the existence of the 12 K-transition. The spectral features present quite an interesting situation.