Johnson-Nyquist noise for a lossy inductor?
You're correct in thinking that the noise can be evaluated like that. The ideal L1 and C1 in the model are noiseless, all the noise originates from R1 and R2. For an inductor these resistors will have quite a low value so their noise contribution is small compared to other elements in a typical LNA circuit.

I am quite sure your formula 

\$\overline{v^2_n} = 4k_BTR1 + 4k_BTR2\$

isn's correct as it doesn't show the frequency dependency caused by L1 and C1 and also noise cannot be added like that as the sources are uncorrelated.

In general experienced LNA designers do not worry about the noise of the inductors as they already know it can be neglected. That is not to say it is not a good exercise to do once.

Usually in a properly designed LNA the active element (NPN, NMOS etc.) will be the most dominant noise contributor.