If I conquer with a unit before equipping blighted battle axe, will it still be dealt damage from axe?

No. Since the equipped unit conquered at any point during your turn, the axe's end-of-turn condition ("if I didn't conquer this turn") is not met, so it stays attached and deals no damage.

The check happens at the end of your turn and asks whether that unit conquered this turn — it doesn't matter whether the gear was equipped before or after the conquest. FAQ 9662 already confirms the unit itself must be the one that conquered (other units' conquests don't count), but any conquest by that same unit during the turn satisfies the condition.