If you expect that the transfer is going to be substantial (hence detectable), it should be possible. Make sure to use a bright FP: GFP or YFP. Good luck!

But how do I specifically tag that one particular protein in the cell?  

Yes right its a good point; its difficult to specifically tag a phosphorylated protein, but not its unphosphorylated counterpart (if I understand correctly your problem). 

One solution that I would maybe investigate is to try to use a mutant with an acidic amino acid in place of the phosphorylated residue to try mimicking the phosphate:

Lets say that you protein is phosphorylated on a serine or a threonine, you could mutate that residue for an aspartate or glutamate and fused this mutated protein to a YFP or GFP and transfect in your cell of interest and see if its transfer to the second cell. A good negative control would then be the Wt or a mutant in which the phosphorylated residue would be mutated for an unphosphorylable amino acid such as an alanine for serine or threonine or phenylalanine for tyrosine. The comparison of the localization of the three variant of your protein of interest could be helpful to adress the question you are raising.