Before you read the walkthrough proper, for the fight with Quiet you need to learn several things about her capabilities. At a distance, she behaves just like the rest of the snipers that you encounter, so after she picks a good vantage point (at first, this is one of the bigger stone objects around), she starts monitoring the area.
After you get within the sight of Quiet, you will be signalled by the white spots on the screen. If, additionally, the crosshair icon appears (e.g. like in the above screenshot). This means that you have walked within her range of fire. In such a situation, you immediately need to leave the area and tale cover (a single shot should not kill you, but this will severely wound you, and your ranking at the end of the mission will drop). During the mission, try to avoid this kind on dangerous moments. First of all, avoid walking into open spaces. Instead, dash between covers and try crawling (you can quickly drop onto the ground, if only you know that Quiet will soon fire again).
Do not count on Quiet's staying put in one spot, for the whole battle, or her changing her position only slightly. Her abilities allow her to change positions very quickly and to jump off high cliffs. She will make use of her abilities each time you attack her, or approach her and she spots you (she is more sensitive to the sounds of the surroundings than the regular opponents).
Note - At the moment at which she leaves her position, Quiet may leave a frag grenade behind. If you spot her trying to leave you such an apple, get away from the site, as quickly as possible.
After Quiet escapes from a given spot, she additionally initiates her camo, which means that you will have to locate her again. Therefore, try to pick the direction in which she escaped, while activating camo. Of course, take your time and do not haste to get to her next position, because you can be targeted again and shot.
You can be helped, while finding out where Quiet hides, by the D-Dog, which is capable of sniffing her out, within the range of several dozens of meters. This will allow you to get prepared properly, before the next confrontation, or to pick an alternative route, thanks to which you will get to a spot appropriate for a sneak attack.