Where did the $Q$ approximation for $RLC$ come from?

I didn't do this explicitly in class- it's a bit lengthy. But the derivation is given in, e.g. Fortney, pp. 64-65. The idea is to write down a damped oscillatory solution to the RLC circuit differential equation for $V(t)$; since energy dissipated is proportional to the square of $V$, you can then plug this in to the definition of $Q$ (ratio of total energy to energy loss per cycle). Using some Taylor series approximations, the expression $Q=\frac{\omega L}{R}$ follows. Near resonance, we replace $\omega$ with $\omega_r$.