True - special relativity predicts the equivalence of mass and energy, and it was formulated before quantum mechanics.
False - conversion of mass to energy and vice-versa is not part of quantum mechanics. The modern formulation of quantum mechanics, quantum field theory, requires matter/antimatter annihilation (mass converted to energy) and pair production (a process in which a sufficiently energetic photon is converted into a particle and an antiparticle - energy to mass conversion). Both are experimentally observed.
I didn't say it wasn't part of Quantum Mechanics. To the extent that the Special Theory is correct, Quantum Mechanics must include relativity, or be mistaken.
[In fact, the Schroedinger Equation does not include it, and is "wrong." It wasn't until Paul A.M. Dirac formulated an explicitly relativistic version of the wave equation that Quantum Mechanics included pair production, negative energy, spin, or antimatter. ALL are relativistic effects.]
The fundamental fact is that mass-energy equivalence is a relativistic effect and is present in classical mechanics (with relativity) and classical electrodynamics; and Quantum Mechanics is not required to have mass/energy conversion in the low energy realm. Lots of solid state physicists, electrical engineers and chemists just use Schroedinger and get along quite well without it. I would refer you to either Bjorken and Drell's Relativistic Quantum Fields or their equally excellent Relativistic Quantum Mechanics, two classic texts which are sitting about two feet away from me as I type for a complete discussion.]