Quantum Mechanics, Concise Book

Practically speaking, QM can be taught without the assumption that students understand the Hamiltonian formalism, simply by starting with Hilbert spaces and operators on Hilbert spaces. In fact, I would claim that having taken a class on basic linear algebra would better prepare you to understand quantum mechanics than mastering classical mechanics. QM is generally taught by referencing classical mechanics, but I believe that's more reflective of the fact that most universities require classical mechanics as a core course, and students coming in to QM will have generally taken it.