Yes, Paladins can use axes in WoW Forever: both one-handed and two-handed axes. They can also use maces, swords and polearms. They can't use daggers, fist weapons, staves, wands or any ranged weapon. This comes from a trainer list compiled during the beta, because Blizzard hasn't published a weapon table for Forever. So we label it Beta data and check it against live trainers.

Weapon Paladin
One-handed axes Yes
Two-handed axes Yes
One- and two-handed maces Yes
One- and two-handed swords Yes
Polearms Yes
Daggers No
Fist weapons No
Staves No
Bows, guns, crossbows, thrown, wands No

Do you have to train it

Weapon skills are learned from weapon masters in the capital cities, as in Classic. If you pick up an axe you can't equip, check that you've trained the skill. Our class weapons guide lists every class.

Which weapon to level with

Retribution wants a slow two-handed weapon, and an axe is as good as a sword or mace if the stats are right. Protection and Holy use a one-hander with a shield. Our Paladin guide and the Paladin leveling plan cover the rest, and Paladin professions has Blacksmithing for weightstones.

Blizzard changed a lot about Paladins in Forever. See everything that changed.