Yes, you can use it as both phase transfer catalyst and supporting electrolyte. But the conductance of this may not be high. Mostly, for organic medium, tertiarybutyl ammonium bromide is used as the electrolyte. 

Just to follow up...tetrabutyl ammonium is a very common cation, but I have seen (and used) perchlorate and hexafluorophosphate as the anion with much greater frequency (TBAP, TBAH).  For more nonpolar organics even larger anions [and cations] will be used to enhance solubility and reduce ion pairing interactions.  

If I were you, I would first run some CVs just with the tertiary alkylammonium salts of your interest, in the specific electrochemical cell that you will utilise. It can turn out that the electrochemical window with your working (or even the counterelectrode if there is no membrane separating them) is small or non-existing in the operational range in which you want to do the benzoquinone reactions.

If your working electrode is e.g. platinum, chances are high that your IL gets easily oxidized or reduced.