If I recall correctly, most of the material in Kittel (or a solid state introductory lecture) is about new concepts, e.g. lattices, reciprocal space, band structure, and doesn't build heavily on quantum mechanics. Of course, there is QM beneath it, but you don't need much rigorous QM. Basically just the wave mechanics part - solving differential equations, finding a wave function for a given potential, etc., and a basic understanding of how quantum particles behave.
For the later chapters, the advanced maths of a QM II course could be helpful if you want to follow the derivations, e.g. calculating complex integrals, residue theorem and so on. But I'm not sure how much of that is actually in the book and how much was in the lectures I heard.
The best would be to just grab the book from a library (or an ebook) and browse through it - you'll notice pretty soon if there are any new concepts/notations you have to learn about first.