The quantum geometric properties of topological materials underpin many exotic physical phenomena and applications. Quantum nonlinearity has emerged as a powerful probe for revealing these properties. The Berry curvature dipole in nonmagnetic materials and the quantum metric dipole in antiferromagnets have been explored by studying the second-order nonlinear Hall effect. Although the quadrupole moment of the quantum geometric tensor is theoretically predicted to induce higher-order quantum nonlinearity, the quantum metric quadrupole remains experimentally unexplored. Here, we report the quantum metric quadrupole induced third-order nonlinear longitudinal electrical response in few-layer WTe_{2}, persisting up to room temperature. Angle-resolved third-harmonic current-voltage characteristics are found consistent with the intrinsic crystal symmetry of WTe_{2}. Through temperature variation and scaling analysis, we identify the quantum metric quadrupole as the physical origin of the observed third-order longitudinal nonlinearity. Additionally, we determine the angle dependence of the quantum metric quadrupole, establishing third-order nonlinearity as an efficient method for revealing the quantum metric structure.