Many intro QM books/courses introduce the appropriate linear-algebra concepts as they come up in the QM material. In that case it's not absolutely necessary to have taken linear algebra beforehand. However, it's definitely helpful, because then you have fewer new concepts to wrap your mind around all at once.
When I was an undergrad, my first QM course introduced the linear algebra as part of the course, but I had already taken linear algebra anyway. I thought it was really cool to see how the stuff I had learned in linear algebra class, which was basically vector and matrix oriented, reappeared in a completely different context in QM.
