Hi, If I defend my quest zone with valkya and my opponent puts 4 damage on her. can I use her action ability to move that damage to one of his units before she dies?

Move apply damage only.

Redirect Assigned damage only.

In other words. She must be alive to trigger her ability and as soon as she has damage tokens on her equal to or greater than her current HP she is immediately discarded from play before you have a chance to take an action such as triggering her ability.