A game is a series of decisions; a good game is a series of interesting decisions
No, you may not use fatigue to move during a guard order attack. In fact, you may not do any movement point based actions during a guard attack. You may only attack. Note however, you may use fatigue to add additional power dice during the guard attack.
While this is true in general, there is a hero in an expansion (Tahlia in Altar of Despair) that has the special ability to move during a Guard order. Note, however, that the base game rules specify that Fatigue may be spent for movement points only during the hero's turn, so the restriction still applies, even to this hero. Also note that there is no such restriction on spending Fatigue for extra power dice.
Descent JitD base Rules p.18 wrote:
Spending Fatigue for Movement
At any time during a hero’s turn, he may spend one
fatigue to gain one movement point, even if he is currently
taking the battle action. This may be done as often
as the hero desires. Movement points gained in this
manner are spent just like normal movement points.
Spending Fatigue for Attacks
After rolling an attack, a hero may spend one fatigue to
roll one additional power die, adding its result to the
roll. This may be done as often as the hero desires, one
die at a time.
Emphasis (underline, not bold) added.