The only 18 hole golf course on Nevis belongs to the Four Seasons Hotel. I was there last week (anchored off Pinney's Beach) and saw the course - it is pristine and beautiful. If you do head to Nevis, I liked the Golden Rock Hotel and it is priced better than the awesome Montpelier Plantation Hotel.

Addendum - the other golf course is a private one, and that isn't a full 18 hole course and I'm not sure it takes greens-fee players.