However I don't know if you can find one long enough if the knot is inside a very long bead. I use carbide bits designed for wood or metal working.

I gently twist the drill bit (using a pin vise to hold it) to cut through the knot.

you mean by hand, or do you attach the drill bit to an actual drill

any metal drill bit in a narrow hole risks fracturing the bead, specially if the hole isn't straight- which is the likely reason its stuck.
i recommend trying pushing with a flexible piece of wire (from the other end) not pulling- and/or using a very skinny broacher by hand, not a twist drill which will be more likely to break off and then the bead is totally ruined...
sometimes the separate filaments of thread will break one by one if you work slowly and patiently
until what remains can slide out.

As Stefany says, applying too much pressure may fracture the bead.

I have used this method especially when the knot is glued or has otherwise become very rigid. The bit has to be very sharp and essentially you are cutting the knot material away gradually.

The pin vise is an inexpensive hand tool that is readily available. I bought several solid carbide, brad point twist drill bits in small sizes for my tool kit and these have worked well for knot removal. The bits are made by ultratool and I buy from McMaster-Carr. They are about $15 each. Very sharp!

Dental picks have a small diameter. It should fit in a bead hole without damaging the bead.

Would something like draino dissolve the silk without hurting the quartz?

No!!!
Do not use draino.
Draino will destroy quartz.

You generally cannot be sure what material is the fibre, and it may be blended, waxed, or coated or fatally impregnated with superglue- so proceed with every possible caution-

I would be inclined to dip a long thin needle or wire into acetone, and insert it into the short end of the perforation (or where the knot is); wait a couple of minures, and push out from the opposite end.

I have had to try most or all of the previous methods, at one time or another. The only time I have broken a bead is when I impatiently tugged too hard—when I should have been pushing from the opposite end. Take your time and work on it bit by bit, and repeat as necessary.

JDA.

Suzi and I use what is called a bead reamer. It has a tapered roughly 2 inch long tip, attached is an image should be able to find them on Ebay very cheap. If not let me know I may be able to help.
Attached is an image tip about 3/4 mm large end of taper about 2mm.
Danny

Well, nothing worked.
Finally, I gave up and glued the ends of two lengths of beading thread into either end of the bead. I've probably f'd up whatever value it had, and I feel like a schmuck, but at least I can use it in this necklace. As long as I'm wearing it against a dark background the problem isn't visible. And I made up some custom silver bead caps to protect the ends.

These started life as cigarette holders, I'm told.

Get a fishing hook..straighten it. Light a candle and heat up the straightened hook. Put it into the drill hole ..the synthetic string will melt and hardened ..glueing itself to the tip of the hook. Problem over.
This is the NORBU'S HOOK ...stuck knots removal methods for beads.
Good luck.