You can separate the labeled protein from free dye using a gel filtration (or size exclusion) column.

Yes following Adam's point, the free dyes and protein conjugates can be separated using either analytical scale size exclusion chromatography (or GPC) techniques coupled with photodiode detectors to quantify the amounts of both portions quantitatively. If you just want to run simple experiment and check qualitatively, even some thin layer chromatogram might work, assuming the amount of unbound dye is sufficient for detection by your eye.

Absorption spectra of coupled and free dye should be different. You can check absorption spectrum of solution if you know the absorption maximum of dye.