<p>The surface states of three-dimensional topological insulators possess geometric structures that imprint distinctive signatures on electronic transport. A prime example is the Berry curvature, which controls, for instance, electric frequency doubling via its higher-order moments. In addition to the Berry curvature, topological surface states are expected to exhibit a quantum metric, which plays a key role in nonlinear magnetotransport. Here we provide evidence for a nonlinear response activated by the quantum metric of the topological surface states of Sb₂Te₃. We measure a time-reversal-odd, nonlinear magnetoresistance that is independent of temperature and scattering time below 30 K, and is thus of intrinsic geometrical origin. This quantum metric magnetoresistance can be controlled by tuning the contributions of the top and bottom topological surface states through voltage gating. Our measurements demonstrate the existence and tunability of quantum geometry-induced transport in topological phases of matter and enable the design of functional topological devices.</p>