They didn't have to kill her though...She was literally just chillin' (<- lol sorry not sorry) in her own little territory well away from civilization without ever coming out nor attacking anyone who did not come to her first. The safe zones were created by destroying the gates scattered around the world. THOSE were the cause of a lot of damage to human civilization since a gate left alone too long would break and spit out a bunch of violent monsters.

The reason they hunted FQ was because they thought she would build up strength and armies and leave her territory to cause damage to humans. She had done nothing before the heroes went to her. In other words, they saw her as a POTENTIAL threat and went to subjugate before she could move. Like how farmers will go out and cull wild pig herds before they get close to their crops. There's no guarantee the boars will come, but just in case they take them down first...kinda reminds me of the anime Psycho-Pass now that I think of it...lol