Da Costa is a Portuguese surname. Never seen it spelled D'Costa.
Research on Google and put D’Costa and you will see it comes up. Good info. It is actually of Italian, Portuguese and Spanish origin. It is one of those where the origin is multiple and as a result the spelling can vary.
I have seen D’Costa a lot. That is why I asked.
Edited to clarify: how the spelling has changed and does not follow the rules of the original language requires research but the last name D’Costa exists.