Series to Parallel inductance conversion
You are confusing the Q of the inductance with the Q of the circuit. In a series LCR circuit, the Q of the inductor is the ratio of the voltage across the inductor to the voltage across the circuit. As you say, this is also the ratio of the inductive reactance to the resistance. In a circuit with series R &amp; L in parallel with C, this relation also holds.

In a parallel LCR circuit, the resistance is no longer a property of the inductor itself which is why you get a different definition for Q. The Q in this case is more a property of the circuit than the inductor. You cannot convert one circuit to the other although you could find two circuits with identical Q.

The Laplace transforms of the impedance for (r + L) // C is ...

$$\frac{\frac{r}{LC}+\mathrm s\frac{1}{C}}{\mathrm s^2+\mathrm s\frac{r}{L}+\frac{1}{LC}}$$

for which Q=\$\frac{1}{r}\sqrt{\frac{L}{C}}\$ (at resonance)

The Laplace transform of the impedance for R // L // C is ...

$$\frac{\mathrm s\frac{1}{C}}{\mathrm s^2+\mathrm s\frac{1}{RC}+\frac{1}{LC}}$$

for which Q=\$R\sqrt{\frac{C}{L}}\$ (at resonance)