If your function generator is stable enough (synthesizer), and you can vary the frequency with fine steps (or it can sweep), it can be done with mtwieg's method. Instead of the VNA, you use the spectrum analyser as RF detector to find the -3dB points.
To make sure you measure the unloaded Q, you need to do a second measurement where you increase the distance between the TX and RX coil and your thick copper coil. If you measure higher Q factor, you need to further increase the distance. with Q factors > 1000, every structure in the vicinity will absorb energy and lower the Q factor (from experience at 14 MHz).
I understand you are going to use two resonators (RX side and TX side) for power transfer (critically coupled BPF?). I don't know the distance between the coils, but once you have the coupling coefficient (you can get that from EM simulation or using approximated formulas), you can put the whole thing in a circuit simulator, saving you lots of time.
If your system has to work over a range of distances, it is still feasible, but becomes more complicated.
Other thing, what type of amplifier do you have? Why this question?. It is very likely that during the experimentation, the amplifier will se very bad loads.
For practical measurements in a 50 Ohms system, I would definitely invest in a return loss bridge, VSWR bridge, etc, to get fast idea on the mismatch. For this frequency you can make these things yourself and test with know resistances. It is standard instrumentation for many hams (radio amateurs).