Can glass be deionized?
You can indeed "harden" glass by exchanging ions on the surface, but not because it turns into quartz glass.

Quartz glass is mostly so robust because it has an extremely small thermal expansion coefficient, and is therefore mostly free of internal stress due to uneven cooling. Ordinary glass already gets a lot more durable when you temper it, thereby relieving those internal stresses.

Generally, glass breaks under a very small extensional strain. It has a huge $E$ modulus and can take a lot of stress, but when you bend it, so the extensional stress and deformation is largest at the surface, where also you always have small scratches etc.), it shatters.

The trick is to put the glass surface under a large compressive stress. That way it can take a lot more pull before the critical strain (deformation) is reached and breakage occurs.

There are basically three ways to do that: Firstly, you can coat the hot glass with another glass that has a lower thermal expansion coefficient. That way the surface gets compressed during cooling. Tricky, and expensive.

Then, the most common way, is to rapidly cool the still nearly molten surface by e.g. blowing cold air on it. The outer shell is already cold, the inner part takes longer to cool and still shrinks, thereby compressing the surface. 

The last, chemical trick (making supremely solid glassware) is to exchange sodium ions in the surface layer (100nm or so i think) by potassium ions. This was done in the eighties in the GDR for household glassware, under the brand name "SUPERFEST", by soaking the glasses in molten potassium nitrate (at 400 °C) for twenty minutes or so. The ions are larger, but the silica network lattice does not yield below at least 500°C, so the surface gets compressed by the exchange.

You know you had a glass that was treated in one of these ways when it does not break into large shards, but basically crumbles into small pieces or even dust. The classic example for the latter is Prince Rupert's drop, of the former any car windshield. (It seems the Na/K exchange does not (necessarily) do that, see your mobile phones "Gorilla glas".)