Surface defects and organic surface-capping ligands affect the photoluminescence properties of semiconductor quantum dots (QDs) by altering the rates of competing nonradiative relaxation processes. In this study, broadband two-dimensional electronic spectroscopy reveals that absorption of light by QDs prepares vibronic excitons, excited states derived from quantum coherent mixing of the core electronic and ligand vibrational states. Rapidly damped coherent wavepacket motions of the ligands are observed during hot-carrier cooling, with vibronic coherence transferred to the photoluminescent state. These findings suggest a many-electron, molecular theory for the electronic structure of QDs, which is supported by calculations of the structures of conical intersections between the exciton potential surfaces of a small ammonia-passivated model CdSe nanoparticle.