Relentless: "When you declare an attack targeting a figure within 3 spaces, that figure suffers 1 Strain"

No Quarter: "When an Imperial figure declares an attack targeting a hero who has suffered 2 or more Strain, apply +1 Surge to the attack results"

If a Trandoshan Hunter declares an attack on a hero that already has 1 Strain, the hero would have 2 Strain during the Declare Attack stage of the attack. Does this trigger No Quarter (i.e. the attack gains +1 Surge)?

On a side note, this seems to be the inverse of the interaction between Oppression ("When a hero who has suffered 2 or more Strain declares an attack, apply +1 Evade to the defense results") and Strain-imposing abilities that are used during the Declare Attack stage (e.g. Diala's Precise Strike). I suppose the ruling is similar for both cases?

Performing an ability seems analogous to paying the cost to perform an ability - it doesn't cause the original trigger to be re-evaluated:

1. Declare a valid target (in line of sight)

if the target is hero who has suffered 2 or more strain, No Quarter triggers
if the target is a figure within 3 spaces, Relentless triggers.

If both No Quarter and Relentless triggered, they triggered at the same time and you can choose the order in which they are resolved.

If only Relentless triggers, it is performed.

If only No Quarter triggers, it is performed.

If neither trigger, neither is performed.

So if the hero had only 1 strain when attack was declared, No Quarter will not trigger, even if Relentless causes the hero to have 2 strain during the rest of the attack?

Right. When the attack was declared, the hero had less than 2 strain.

It would be different if No Quarter was instead: "When an Imperial figure declares an attack, if the target is a hero who has suffered 2 or more Strain, apply +1 Surge to the attack results", the trigger would be "when declaring an attack" and the ability would always trigger on imperial attacks, and the target and strain would be checked when the ability is resolved.

What about Prey Upon Doubt and Oppression? i.e. if the Rebel hero is at 1 Strain, Imperial player exhausts Prey Upon Doubt and Rebel player chooses to take the strain, would this trigger Oppression?

No, Prey Upon Doubt + Oppression is completely analogous to the Precise Strike + Oppression case.