For sorting unknown caps, the measurement procedure is at least an LCR meter to get a ballpark figure or a network analyzer/impedance analyzer to actually characterize it... not generally the cheapest of options. Either device could give you a Q factor reading for the frequency it tests at, just generally if it's over a couple of MHz an LCR meter simply isn't going to have the frequency range.
With a single measurement frequency you can probably get an idea of the dielectric by matching up with some manufacturer example curves for the same value/size cap, but with a few frequencies, ideally nearish to where they'll be operating, you can take some points and try to fit it to an example curve, which, while relying on a lot of assumptions, should get you close in many cases... but at that point it's probably less effort/expense to just buy some caps that you have a datasheet for.
If you've got a sig gen that can produce the frequency you're interested in and an AC voltmeter with that bandwidth, you could try that way with a little math. Since Q factor isn't really concerned with phase, as the network analyzer or LCR meter is, you can measure the sig gen through the shorted test fixture to the meter, then the same measurement with the capacitor in the fixture. The difference should be the dissipation of the capacitor (since the inductance of the jig has been measured with the shorted measurement), and your Q factor is 1/dissipation factor. Still need a fair bit of bandwidth on your voltmeter if you're looking for higher frequency stuff, but not looking at phase information means much less expensive equipment requirements.