We investigate the terahertz (THz)-pulse-driven nonlinear response in the d-wave cuprate superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+x} (Bi2212) using a THz pump near-infrared probe scheme in the time domain. We observe an oscillatory behavior of the optical reflectivity that follows the THz electric field squared and is markedly enhanced below T_{c}. The corresponding third-order nonlinear effect exhibits both A_{1g} and B_{1g} symmetry components, which are decomposed from polarization-resolved measurements. A comparison with a BCS calculation of the nonlinear susceptibility indicates that the A_{1g} component is associated with the Higgs mode of the d-wave order parameter.