How can you find Quality Factor of an arbitrary RLC Circuit?
If you want to determine the impedance \$Z_{in}(s)\$ of this network, you can apply the fast analytical circuits techniques or FACTs described in the book I wrote. To do so, the stimulus you need to apply the network across the network is reduced to 0 A and you "look" across each energy-storing components to express the resistance you "see":



The zero is determined by nulling the response when the excitation is back. It means determining the impedance condition in this circuit for which \$V_T\$ becomes 0 V. And this is when the inductance and its series resistance form a transformed short circuit leading to: \$\omega_z=\frac{R_1}{L_2}\$.

Once you have determined all these elements, a quick Mathcad sheet lets you test the raw expression versus the one determined using the FACTs. The quality factor \$Q\$ is determined by identifying the factors in the denominator with those of the formalized 2nd-order polynomial:

 

If we consider \$R_1\$ fairly small compared to \$R_2\$, we can write \$Q=\frac{R_2\sqrt{C_1L_2}}{L_2+C_1R_1R_2}\$.