We have investigated the optical properties of a single InAsP quantum dot embedded in a standing InP nanowire. A regular array of nanowires was fabricated by epitaxial growth and electron-beam patterning. The elongation of transverse exciton spin relaxation time of the exciton state with decreasing excitation power was observed by first-order photon correlation measurements. This behavior is well explained by the motional narrowing mechanism induced by Gaussian fluctuations of environmental charges in the InP nanowire. The longitudinal exciton spin relaxation time was evaluated by the degree of the random polarization of emission originating from exciton state confined in a single nanowire quantum dots by using Mueller Calculus based on Stokes parameters representation.