Quantum mechanics is not derived from classical mechanics or energy conservation, but there are "jumping off points" in classical mechanics that may serve to answer your question.
If you study classical mechanics at a sufficiently advanced level you will discover the Hamiltonian formalism. The Hamiltonian for an isolated system with only conservative interactions is the energy and it is conserved. In such systems it is a function of the generalized coordinates (q) and their time derivatives ($H(q, \frac{dq}{dt})$). In the Hamiltonian formalism there are entities called Poisson brackets (I'll leave it to you to look up their definition but they are represented like this [a,b]). Once you understand Poisson brackets, you can show that the equations of motion for your system are: $$\frac{dq_i}{dt}=[q_i,H]$$ where $q_i$ is one of the generalized coordinates.
Now let's jump from classical mechanics to quantum mechanics. The Planck hypothesis was that energies are no longer continuous but discrete (always proportional to the Planck constant h). He was forced to introduce this assumption in order to derive the black body radiation curve. Heisenberg realized that he could preserve most of the classical formalism if he introduced operators for the classical variables and associated their Poisson bracket with the operator commutator like this: $$[a,b]->\frac{2\pi}{ih}(ab-ba).$$ This led to the uncertainty principle and the matrix formulation of quantum mechanics.
Schrodinger realized that he could form a differential equation for something called a wave function by substituting Heisenberg's operators into the classical Hamiltonian function. It took a while for the probability interpretation of the wave function to jell and to prove that the Schrodinger equation approach was completely equivalent to Heisenberg's matrix mechanics, but that in a nutshell is how quantum mechanics was born (pun intended). Lest I be down voted for slighting Bohr and the Old Quantum Theory (OQT), I should add that I consider OQT the gestation (not the birth) of quantum theory.