Note: If you want 100% completion, you’ll have to let her go. If you prevent her from leaving, it will keep you from doing a mission that leads to the true ending.
Supposedly, there are two ways to stop Quiet from leaving your side:
Lower your bond level – if your bond level isn’t at 100%, she won’t leave. However, this is hard to maintain, and locks out certain abilities of hers.
Put the Butterfly (front) design on your emblem before you finish Mission 41 – this is the option you should go for.
We haven’t been able to confirm this ourselves, as we haven’t reached the mission in question, but many people have said this worked for them.