The electron transport property of a graphene nanoribbon （GNR） quantum dot in a dilution refrigerator at temperature 20 mK is studied. The Coulomb blockade diamonds and excited energy levels of the quantum dot are clearly observed. It is found that the statistical distribution of the spacing between the nearest-neighboring Coulomb blockade peaks and that of the individual peak height are consistent respectively with the typical Wigner-Dyson distribution and Porter-Thomas distribution as prescribed by the random matrix theory for a quantum chaotic system. Thus， our results demonstrate that the GNR quantum dot is a quantum chaotic system at low temperatures. The possible origin of this phenomenon is also discussed.