I believe (but didn't check in length) that any reasonable cooling speed quenches ordinary glass. Thick glass has been made for millenniums, so quenching fibres is trivial.
Glass is formed at a high temperature, but internal stresses build up during cooling through a much lower temperature range, where the material has already some strength. This is typically the annealing temperature. I don't believe (not checked) that annealing brings something before quenching. It makes a difference for alloys because of segregation; they are first "solution-treated".
Quenching creates a compressive stress in the skin.
The chemical composition of glass varies a lot.