Zero and longitudinal field mu SR measurements on diluted triangular-lattice antiferromagnet LuFeMgO4 have revealed gradual and highly anisotropic slowing-down of spin fluctuation. Relaxation rate lambda in the condition with initial muon spins perpendicular to the hexagonal c-plane shows divergent behavior while that in the parallel configuration remains finite. At certain temperature range the fluctuation of spins is suppressed by a small external magnetic field. Monte Carlo calculation suggests that the correlation time for z(c)-component of spins diverges first on cooling.