Carbon is one of the most versatile elements which exists in various forms. In this question, we are asked to identify the form of carbon that does not actually contain carbon.

Option 1: Graphene is a form of carbon which is made up of a single layer of carbon atoms arranged in a hexagonal lattice. It is a two-dimensional material with extraordinary properties.

Option 2: Graphite is also a form of carbon, composed of stacked layers of carbon atoms arranged in a hexagonal lattice. It is commonly used as the "lead" in pencils due to its slippery nature.

Option 3: Fullerene is another form of carbon consisting of a large molecule made up of carbon atoms arranged in a spherical or cylindrical structure. It is also known as "buckyballs" or "carbon nanotubes".

Option 4: Quartz, on the other hand, is not a form of carbon. It is actually a mineral composed of silicon and oxygen atoms. While it may appear similar to some carbon-based compounds such as diamond, it does not contain carbon itself.

Therefore, the correct answer is option 4 – Quartz.