I treppan telescope mirrors on a regular basis often using an unmodified hole saw in a drill press, 40 grit abrasive and a lot of water. It's a bit messy but it works every time and I've yet to damage a mirror. With large custom tools like you see here, I prefer the stability of a vertical mill.
With smaller holes the technique is the same and you can use a drill press.
Here is a smaller mirror being trepanned from a larger disk with a hole saw.
I have used this method for cutting glass as thick as two inches in less than ten minutes. Often the mirror is a carefully figured parabola worth well into four figures.
Buy a suitable length of steel or iron pipe with the appropriate OD, chuck it in the drill press. If you can't find 40 grit carborundum, you can substitute a valve grinding compound. Do NOT apply any excessive pressure! Let the abrasive do the work.
Of paramount importance is to test that the glass is NOT tempered as has been suggested earlier.
I discourage the use of drill bits. carbide or otherwise. They are too aggressive and if they should grab the material, the game is over!
"Both the revolutionary and the creative individual are perpetual juveniles. The revolutionary does not grow up because he cannot grow, while the creative individual cannot grow up because he keeps growing." Eric Hoffer