SPOILER WARNING: Story spoilers are below.
Quiet is arguably the most potent buddy in Metal Gear Solid V: The Phantom Pain. Unfortunately, after Mission 43 she departs. This results in you not having access to her during post-game gameplay. Thankfully, there is a way to keep her as a buddy even after the events of Mission 43 and 45.
The trick is to equip the Butterfly (front) design to your emblem prior to Mission 41.
Before you can equip the Butterfly design, you will need to acquire it. It is a reward for you earning codename Butterfly for 3 missions in a row. This codename is granted whenever Quiet earns more kills than you in a mission, and you are spotted. Which mission(s) you choose to complete to earn these codenames is up to you (you can repeat older missions), but you can't acquire the Butterfly design through Side Ops.
Once you've unlocked the Butterfly design, equip it to the front of your emblem prior to Mission 41. Quiet will still depart, but you will have access to her after the fact.
Important note: equipping the Butterfly design on your emblem even once will render you ineligible for unlocking Mission 46.