No, Chris Martin has not had a hair transplant.
You can tell by comparing photos of him from a few years ago with what he looks like now...
Back when he first got with Gwyneth Paltrow, he certainly had a bit of a receding hairline:
People then started to get suspicious about Chris Martin having a hair transplant when he appeared performing with his band Coldplay: Suddenly his hair looked thick, curly and bushy... A full head of hair!
But, fast-forward a few months, and he's had a haircut that exposes his natural hairline, and you can plainly see the same receding pattern he was known for.
So it's not a case of him getting a transplant. Rather, the hair he did have was allowed to grow thicker, giving the impression that his hairline had shifted.