Depends on a couple of things, How long you've been in NB, why you are here, etc.
Either pickup a copy of the Hunting/Trapping Summary or view it here
http://www.gnb.ca/0078/publications/HuntTrap-e.pdf#pagemode=bookmarks
Pretty much most of what you need to know is in there.
You will also need to show proof of passing an approved Hunter's Safety course. You may be able to use an old hunting license in lieu of a course but you'll have to verify that with the DNR as I do not know if they will accept an out of province license.
FWIW, I have never been asked if I had a firearms license when I buy my license.
A Class 3 (resident deer/small game License) cost $38.42.