There are two ways, in which you can win with Quiet - by depriving her of her entire, red, health bar (do not worry, because it is impossible to kill her prematurely this way), or by depleting her whole, blue, endurance bar. There are not any long-term effects to the decision that you make, because either way, after you have sufficiently weakened Quiet, you will be able to sneak up on her and decide her fate (this has been discussed on the next pages of this walkthrough).
If you want to win the battle by depleting Quiet's endurance bar, your best weapon is the sniper rifle. In the situation, in which Quiet learns your position, stay behind your cover and lean out only to take the shot. Then, walk away quickly, or else her attack ends in a success. If, on the other hand, she does not know your position, you can risk a more precise shot at the head, of course.
There are many other ways to deplete Quiet's health bar. For example, you can use the rocket launcher or the grenade launcher. If you managed to get close to her current cover, you can surprise her with a grenade and hope that it does not go off before she escapes. Using standard weapons, e.g. a machine gun, is not recommended, because you don't deal too much damage to her, this way, because she escapes into another spot.
If you want to win the battle by depleting her endurance bar, you can use weapons with incapacitating and sleeping ammo. While playing this mission for the first time, you, unfortunately, your only option will probably be a pistol and sleeping ammo. Due to this, you will have to sneak up on Quiet and, additionally, your attacks will only deal a little bit of damage to her. While replaying this mission, later into the campaign, it is a good idea to use some better weapon, e.g. a sniper rifle with sleeping ammo.
The best way to deprive Quiet of her endurance bar is to...deploy supply drops onto her. If only you have enough cash, pick Supply Drop via the iDroid and set Quiet's current cover as the destination. An example screen of Supply Drop request has been presented in the above screenshot - the selected deployment spot coincides with Quiet's current cover.
After you have requested a supply drop, stay behind your cover and do not lean out. Also, do not come close to Quiet, or else you will drive her away. If you have been following the plan, the cargo should drop onto her and deprive her of more than a half of her health bar. To defeat Quiet this way, therefore, you require two supply drops, at least, but she should not get resistant to this, after the first successful drop.
Note - An additional upside of cargo drops is that this why, you complete two secondary objectives of this mission (in the meantime, do not attack her with any weapons).