No. Protect won't change it's form, nor will any other status move. Only King's Shield can trigger Stance Change.
It changes to Blade Forme when using a damaging move, and reverts to Shield Forme when using King's Shield. Status moves other than King's Shield will not change its form. Moves that call other moves, such as Sleep Talk, will not change Aegislash's Forme, even if they call a damaging move or King's Shield.