Armored Tusker of enemy moved from his base to my controlled battlefield "Fortified Position". Will his Armored Tusker be able to deal damage first equal to his shield on his move trigger before I can trigger the "Fortified Position" battlefield trigger "when I defend" where I gain shield +2 to unit?

Yes — Armored Tusker's "When I move" trigger resolves before Fortified Position's defend trigger even exists, so you can't have Shield 2 up in time to blunt that damage.

The move trigger goes on the Chain in the cleanup right after the move, and a Showdown can't open while the Chain has items (Rules 344, 462), so it resolves before Combat starts — meaning before Attacker/Defender designations are handed out. Fortified Position only triggers when you actually gain the Defender designation at the start of the Combat Showdown (Rule 383.4.f), which is after the Tusker's damage has already been dealt and resolved. Shield 2 is also "this combat" and only ever a +2 [M] while defending, so it would never have applied to pre-combat damage anyway.

One correction on the framing: the Tusker is the attacker here, not a defender, but it still deals the full amount — "damage equal to my [Shield]" reads the Shield Value as a characteristic (Rule 814.3), so it's 3 damage regardless of designation. Your Shield 2 does matter from then on, for the Combat Damage Step, so pick the unit you intend to actually defend with.