Transmission electron microscopy would be a good way to show that the carbon dots were present. Organic molecules would not be visible, but carbon dots would show up as little spherical particles. Dynamic light scattering would also show carbon dots as particles of a few nanometers in size, whereas organic fluorophores would not be detected (unless very concentrated, and then much smaller). Size exclusion chromatography would be a good way to remove any organic fluorophores, since the carbon dots would elute much earlier than the much smaller organic fluorophores.
This paper should suggest several additional methods of characterization.