That's quite a strange question. Can't say that we see that very often in cats.
My first thought is that this situation is related in some way to the diabetes. Whether this is a primary effect of the disease on the production of the keratin in the nails or some infection (bacterial or fungal) secondary to the diabetes is not clear.
I think it would be wise to do some cytology of the nail beds. If this is bacterial or fungal, a cytology may be able to identify the organisms. A cytology is done by taking some of the material from the source (in this case the nails and nail beds) and smearing it onto a glass slide and evaluating it under a microscope. It's possible that a culture may need to be done if cytology is not definitive. These tests would be able to clarify if medications are indicated and which ones. Antibiotics would be used if the problem was bacterial. Antifungals would be used if it was fungal.
If neither is identified, it may be related to the hormonal and metabolic effects of the diabetes itself. Then it would be vital to get the diabetes very well controlled in order for the secondary issues to resolve.
Hope these thoughts help.