We performed $^{31}$P-NMR measurements on LaFe(As$_{1-x}$P$_{x}$)O to investigate the relationship between antiferromagnetism and superconductivity. The antiferromagnetic (AFM) ordering temperature $T_{rm N}$ and the moment $mu_{rm ord}$ are continuously suppressed with increasing P content $x$ and disappear at $x = 0.3$ where bulk superconductivity appears. At this superconducting $x = 0.3$, quantum critical AFM fluctuations are observed, indicative of the intimate relationship between superconductivity and low-energy AFM fluctuations associated with the quantum-critical point in LaFe(As$_{1-x}$P$_{x}$)O. The relationship is similar to those observed in other isovalent-substitution systems, e.g., BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$ and SrFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$, with the 122 structure. Moreover, the AFM order reappears with further P substitution ($x > 0.4$). The variation of the ground state with respect to the P substitution is considered to be linked to the change in the band character of Fe-3$d$ orbitals around the Fermi level.
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