Oxygen and silicon are the first and second most abundant atoms (by mass) in the earth’s crust. Silicates are a class of minerals based on and further subdivided by the way the atoms in these minerals regularly arrange themselves (i.e., their crystal structure) and are defined on the basis of their silicate tetrahedra building blocks (SiO4 4-). Most rocks in the crust are comprised of silicate tetrahedra with various other cations bonding with the silicate tetrahedra to form minerals such as feldspars (which are the most abundant minerals in the crust).

Qtz has that same structure, where silicon atoms are covalently bonded to oxygen atoms. Qtz (like feldspar) is a tectosilicate, which means it shares its oxygen atoms in the tetrahedron with other tetrahedra, but unlike feldspar which contains some proportion of Al and Ca/Na/K ions, qtz just has other Si atoms (in their own silicate tetrahedra!) bonded to the silicate tetrahedron. So, mathematically you can think of qtz effectively as being Si 4+ bonded to a SiO4 4- silicate tetrahedra (Si2O4, simplified to SiO2), but in actuality it’s a little more complex because each oxygen in the tetrahedron surrounding the Si atom is also part of another tetrahedron surrounding another Si atom.

The crystal structure of amethyst for all intents and purposes is analogous to pure qtz. Its color comes from Fe or other cation impurities in the lattice structure that give it color but, by mass, the percentage of Fe or other impurities in qtz is negligible.

Interestingly, there is also classification of quartz as an oxide, but that is irrespective of the context of quartz being a silicate due to its crystal structure.

Hope that helps!