Making a glass object glow in a colour that it isn't

Just be aware - lighting a scene using an emitter inside a refractive object will be very noisy - as you are essentially lighting the scene using caustics.

That said - what you are after is possible with a bit of light path trickery.

I have two spheres in this scene. A large green glass one and a smaller one inside it with a red emitter material.

These screenshots show the material nodes for the two spheres.