Do tank circuits have Johnson-Nyquist noise?

[jmw]

If I have a parallel L-C circuit with some finite Q, it could be modeled as has having a resistor R in parallel, where \$Q = R \sqrt{C/L}\$. This isn't a "real" physical resistor, so does a L-C circuit have Johnson-Nyquist noise that depends on the Q?

[TimFox]

Yes, when the equivalent circuit has a resistor (series or parallel), that resistor looks like a noise-free resistance in series with Johnson noise voltage generator (or in parallel with a noise current generator) at that location in the circuit.  One or the other, not both.
For narrow frequency bands, you can lump all the circuit losses into one such resistor.
You may need to use more than one resistor for wider band calculations, where the noise generators are statistically independent.
This is how Spice calculates such circuits.
Obviously, calculating the noise at the designated output of the circuit will have a frequency dependence from the tuned circuit.
For wider frequency band, depending on how that resistance was calculated, the resistance itself may be frequency dependent.
Decades ago, I faced exactly this situation with resonant sensor coils in MRI systems, where the bandwidth of the receiver system was narrow, centered on the sensor resonance, and the conductivity of the patient was an important part of the sensor Q.

[Alex Nikitin]

Yes, and it gets complicated when the added losses are frequency and power dependent. One close to my heart example is a magnetic head in a tape recorder. The magnetic energy losses in the core are equivalent to the increase of the coil active part of the impedance (=resistive part). A coil in a magnetic playback head for a tape recorder can have DC resistance about 100-300 Ohm however if you measure the active part of the impedance at AC, it will increase with frequency and the speed of that increase will depend on the core material of the head. For a typical permalloy core the active part of the impedance can easily reach several kOhm at 10kHz. And that means an increased noise produced by the head (which is not a good thing, obviously, in a PB head). Below is the graph I've made some years ago, comparing losses v frequency for 6 PB heads with cores made from different materials. 

Cheers

Alex