Sadly enough, Quistis doesn't get as much screen time as most of the other leading characters in the game, but she did have a very interesting story behind her cool exterior and life as a SeeD. Here's what we know about her, during FFVIII, before and after.
Little to nothing is known about Quistis' earliest part of her life. Her parents are never mentioned, but I assume that they died during the Sorceress War almost two decades before the start of FFVIII. Whether or not she was immediately moved into Edea’s care or if she spent some time elsewhere remains to be seen.
A short time later, Quistis was taken under the care of a young woman by the name of Edea Kramer, a woman who had inherited sorceress powers at a young age and had no direct children of her own. Quistis, along with most of the other characters in the game that had ended up there for various reasons that were probably the result of the bloody war that orphaned them, (Squall, Irvine, Zell, Selphie, Ellone, and Seifer), were taken care of and raised by Edea in a cottage that was located by a light house near the sea on the Centra continent.
Through flashbacks in the game, you learn more about the group's antics as children. Quistis was always the bossy and proper one who earned the nickname from Squall as "Bossy Quisty". Quistis and the others remember how Quistis was always trying to be the boss and take care of everyone else, especially after Ellone left the orphanage to leave for Esthar.
Some time after, Quistis was adopted into an unknown family. (Quistis only mentions this event once and doesn't go into explicit details). Things didn't work out too well for her and by the age of 10 she was transferred as a young student to Balamb Garden Military Academy. She was trained using Guardian Forces which eventually cause you to lose your memories. Quistis' memories were then mostly lost and she forgot all about her childhood in the lighthouse by the sea and of her memories of all her friends vanished.
Years later, both Seifer and Squall both ended up at Balamb Garden as well. (Probably from Edea as Edea is overtaken by Ultimecia's powers from the future. She might have meant this to protect her "children".) Quistis doesn't directly remember them, but found them interested and always tried to mediate between their quarrels. She had a strange attraction towards both of them, mostly Squall, but she didn't know why. She assumed it was love, but later on it's shown that it was just her childhood friendship that drew her to him.
Quistis quickly become a child prodigy in the Garden. By the age of fifteen, she became a SeeD (this was very rare occurance). At seventeen she got her teacher's permit and became an instructor at the Garden. She still had her sights on the two young men and she thought for sure that she was in love though she never let her feelings towards
A year later, the SeeD testing started up and Squall registered himself for the testing after completing the written exams. Quistis joins as his aid during the testing and she is happily surprised that he passes and becomes a SeeD as she always had high expectations for him. (SeeD are special military forces dispatched all over the world for protecting the world against the sorceress and as mercenaries for private companies, ext.) Sadly, after the graduation ball, Quistis finds out that her Instructor's permit will be removed as they don't see her as fit to be an Instructor. Quistis then becomes a full-time SeeD and is no longer Squall's Instructor, but rather a teammate.
Quistis confides to Squall later on how they were the same now and as equals. She was no longer his teacher and she tells him how they might actually have the same line of work someday. Squall gives her the cold shoulder and walks off, obviously uninterested. Quistis feels heartbroken at this, and starts to doubt herself as anything but a failure.
Heading off, Squall goes out to Timber to help the rebellious group known as the Timber Owls headed by the somewhat stubborn and free-spirited Rinoa Heartilly. Quistis sees that Squall has an attraction for the beauty and she decides to give up on ever hoping that Squall would someday love her. Sadly, Quistis sort of fades into the background afterwards and the story centers around Squall and Rinoa's growing relationship.
Time passes, (I'd have to say about half a year or so during the course of the game) and eventually, the group starts to get their memories back. Irvine realizes that the happy-go-lucky Selphie Tilmitt is his "Sephie" from childhood. Afterwards, all the group slowly uncovers their past and are happy that they've found each other. Quistis then realizes that she wasn't in love with Squall; it was simply her "sister-like" instinct that forced her to act that way. She tells Squall this dismissing her attraction to him as "A... misunderstood... love?" (hence the name of this site) I for one, SERIOUSLY doubt that she was telling the truth. I believe she does love Squall, but realizes that Rinoa has his heart and thus decided to move back into the backdrop and let Rinoa and Squall find their happiness because she was very mature and understanding for her age.
After the final battle with Ultimecia, Quistis returns back to the Garden with your team. I believe that after defeating the Sorceress, she becomes an Instructor again since her talents would be very beneficial towards the Garden. Then again, if the Sorceress' powers have been truly defeated, there would no longer be a reason for the Gardens. However, I doubt that could be the case because the soceress' powers are inherited and it's shown that once again, Edea inherits Ultimecia's power, so basically this timeline repeats in an indefinite loop. Great. All that can be said is that in the end, Quistis looks happy.