G-quadruplexes (G4s) formed by the human telomeric sequence AG3 (TTAG3)3 (Tel22) play a key role in cancer and aging. We combined elastic incoherent neutron scattering (EINS) and quasielastic incoherent neutron scattering (QENS) to characterize the internal dynamics of Tel22 G4s and to assess how it is affected by complexation with two standard ligands, Berberine and BRACO19. We show that the interaction with the two ligands induces an increase of the overall mobility of Tel22 as quantified by the mean squared displacements (MSD) of hydrogen atoms. At the same time, the complexes display a lower stiffness than G4 alone. Two different types of motion characterize the G4 nanosecond timescale dynamics. Upon complexation, an increasing fraction of G4 atomic groups participate in this fast dynamics, along with an increase in the relevant characteristic length scales. We suggest that the entropic contribution to the conformational free energy of these motions might be crucial for the complexation mechanisms.