Quiet becomes available as a buddy a long while after mission 11. Technically you can miss your opportunity to gain her as a buddy by choosing the 'wrong' choice at the end of the mission. If you made the wrong choice the first time around, you can replay mission 11 and make the right choice to correct this and gain Quiet as a Diamond Dogs member.
*** Spoiler - click to reveal ***At the end of mission 11, you'll be asked to either kill Quiet by pressing 
or to wait and spare her. Simply wait and don't push any buttons to extract her via the chopper that will pick you up at a nearby landing zone to meet the requirements of this trophy.