Hi fiveworlds. I can probably help.
The Czochralski process is a technique for growing crystals, including silicon crystals. It involves taking the material you wish to grow and melting it in a big pot, called a crucible. You then take a seed crystal (a tiny single crystal oriented in the crystallographic direction you want) and dip it down into the molten material. The seed will be attached to some sort of rod or other support, which is often cooled (for example, by pumping cooling water through it). You very slowly pull the seed crystal up from the melt. A tiny amount of the molten material will solidify (crystallize) on the seed. You slowly do this and grow your crystal.
As far as I know, the only use of quartz in this process is as the crucible material when growing silicon crystals. Quartz is silicon oxide, and so it isn't a serious contamination if some gets into the molten silicon. It is also very heat and corrosion resistant.
But I also don't think they use quartz crystals for that - in fact, what they use I wouldn't call "quartz", though a lot of people do. What they use is a non-crystalline, amorphous silicon oxide glass, that a lot of people call "quartz glass". But technically, quartz is crystalline, glasses are not.
I lot of Czochralski growth uses platinum crucibles; I don't know if they also use those in silicon growth.
By the way, quartz crystals are grown by a completely different process, called hydrothermal growth. It is essentially a solution growth method, grown in water, but the water is typically at about 350C, and to keep it liquid (not steam), you have to do it in a pressure vessel at about 800 to 1000 atmospheres pressure (I used to work for a company that grew quartz crystals).
Let me know if you have any other questions.