The first direct experimental observation of an electric quadrupole ($textit{E}$2) transition between bound states of an atomic negative ion has been made. The transition was observed in the negative ion of bismuth by resonant (1+1) photodetachment from Bi$^-$ $^3textit{P}_2$ via excitation of the Bi$^-$ $^3textit{P}_0$ fine structure state. The $textit{E}$2 transition properties were independently calculated using a hybrid theoretical approach to account for the strong multi-level electron interactions and relativistic effects. The experimental and theoretical results are in excellent agreement, providing valuable new insights into this complex system and forbidden transitions in negative ions.