We have performed (31)P-NMR measurements on Ce(Ru(1-x)Fe(x))PO in order to investigate ferromagnetic (FM) quantum criticality, since a heavy-fermion (HF) ferromagnet CeRuPO with a two-dimensional structure turns into a HF paramagnet by an isovalent Fe substitution for Ru. We found that Ce(Ru(0.15)Fe(0.85))PO shows critical fluctuations down to ~0.3 K, as well as the continuous suppression of Curie temperature and the ordered moments by the Fe substitution. These experimental results suggest the presence of a FM quantum critical point (QCP) at x~0.86, which is a rare example among itinerant ferromagnets. In addition, we point out that the critical behaviors in Ce(Ru(0.15)Fe(0.85))PO share a similarity with those in YbRh(2)Si(2), where the local criticality of f electrons has been discussed. We reveal that Ce(Ru(1-x)Fe(x))PO is a new system to study FM quantum criticality in HF compounds.