We have performed P-31-NMR measurements of the c-axis-aligned poly-crystal Ce(Ru1-xFex)PO with a two-dimensional layered structure in order to understand the origin of T-Curie suppression by Fe substitution. The variation in the magnetic-fluctuation character with respect to x is investigated from the in-plane and out-of-plane fluctuations and the relationship between the static spin susceptibility and the in-plane fluctuation. It was found that three-dimensional ferromagnetic (FM) correlations are dominant and give rise to the FM ordering in CeRuPO. The out-of plane fluctuations are significantly suppressed with increasing x, and it was revealed that the FM fluctuations become two dimensional near a FM quantum critical point (QCP). Our NMR results strongly suggest that a unique FM QCP in Ce(Ru1-xFex)PO is induced by the suppression of the magnetic correlations along the c-axis, which is a different mechanism from that presumed in heavy-fermion compounds.