Page 6 of NB Rules and regs
http://www2.gnb.ca/content/dam/gnb/Departments/nr-rn/pdf/en/Wildlife/HuntTrap.pdf
• The holder of a Varmint Licence or a Bird & Small Game Licence shall only hunt with: º A rim-fire or centre-fire rifle of less than .23 calibre.ºOr: A muzzle-loading firearm. ºOr: A shotgun with lead shot no larger than size BB or steel shot no larger than size F. ºOr: A bow or crossbow.
Don't think the pellet gun would be legal, but not completely sureD