Final Fantasy VIII Guides and Walkthroughs
This quest has two parts, the first part is completely optional and serves no purpose aside from additional dialogue and backstory. If all you're after is the Combat King 003, then all you have to do is the second half of this quest and you can completely ignore the first part. Scroll down the page for what you're looking for.
As for the first part of Zell's Love Quest... It takes place over the first three discs of the game at all different times. Basically, throughout the game at mostly the most annoying times you can think of, you can return to the library for a scene involved in this quest.
Zell has to be in your party every time that he is available to be otherwise you won't get a scene... However there are a few scenes you can see when you're just in control of Squall or only in control of Irvine. The girl in question is in the Balamb Garden Library, shown in the two screenshots above and below these paragraphs.
When you're ready to start the final part of this quest on Disc 3 you'll want to go to Balamb Town and speak with the girl in the red skirt at the entrance. For this part you'll need Zell in your party and you may also have to walk in and out of town a few times while you're at it. The girl in the red skirt doesn't always appear and she doesn't stay on the screen long when she does appear - so don't use turbo speed!
After you speak with her go to Zell's House and there will be a scene with his mother. Once that's finished go to the Inn and spend a night there, in the morning there will be another scene with Zell and girl. When he joins your party again you'll receive Combat King 003.