Conduction electrons interact with lattice vibrations, leading to coupled electron-phonon states. This effect is of fundamental importance in understanding electron transport and energy relaxation in nanoscale systems. We report quantitative results on the electron-phonon coupling strength in CdSe quantum dots (QDs) and rods, obtained by low-temperature scanning tunneling microscopy. We resolve the polaron eigenstates arising from coupling of the electrons to longitudinal optical phonons. The electron-phonon coupling strength is dependent on the electron orbital symmetry, the number of added electrons, and the size and dielectric environment of the QD.