When a photon interacts with an atom three things can happen:
elastic scattering, the photon keeps its energy and changes angle
inelastic scattering, the photon gives part of its energy to the atom and changes angle
absorption, the photon gives all its energy to the atom
In your case it is a common misconception to think that UV light will be all absorbed by the glass.
When glass reflects visible light, like a mirror image, that is 1., elastic scattering.
In your case, when UV light or IR light (non visible) enters the glass deeper into the material, that is 2., inelastic scattering, or 3. absoption. Inelastic scattering will happen as the photons give their energy to the molecules in the glass lattice and transform into the vibrational energy of the molecules (heat the glass).
Absorption can happen too with glass, and will too heat up the material.
The question is only the ratio of the three interactions, but all will happen. With glass,
visible light will mostly be reflected (1. elastic scattering) or will be refracted (will travel through the glass)
non-visible light will mostly be inelastically scattered deeper in the material, and some will be absorbed heating up the material.
In your case, UV light will mostly be inelastically scattered or absorbed by the glass, heating it up.
Now the UV light that glass emits, is depending on the lattice structure of the glass, and the vibrational motions (heat energy) of the molecules in the lattice. Glass can emit UV light spontaneously too, because of the structure of the lattice, but it can do it too because the absorbed light will be re-emitted at a different frequency, glass can emit UV even if no UV is absorbed.