I used to do stained glass, and to my knowledge what you want to do is not possible.
When glass is cut, it will always fracture along a straight line. That means that if you try to drill a hole, a crack will form in a line across where you want the hole to be.
It is possible in stained glass to cut a circle, but to my knowlege you can only do it by *breaking* away the pieces of glass around the *outside* of the circle. (i.e., breaking away pieces formed by straight line cracks tangential to the circle) YOu end up with a circle and a bunch of broken glass bits.
If the globe is made of some other material, like some form of plastic, then drilling a hole would be possible.
Have you thought about using some type of super-strength glue? I am not sure what you are trying to do, but if you wanted to drill holes so that some kind of metal hardware could go through it, you could glue the hardware on instead.