You're right about four figuring into the meaning of quadroon. A quadroon is a person who has one white parent and one mulatto parent. (Mulatto was once widely used to describe a person with one black and one white parent.) The ancestral mix adds up mathematically to a bloodline of one-fourth black.
The word octoroon applies similarly: The offspring of a quadroon and a white claims a lineage of one-eighth black.
Designations of this sort mattered greatly to Europeans colonizing the Americas. Evidence of European heritage established social standing. A clear formula that characterized a person's pedigree may have sounded good on paper. In fact, skin color typically defined racial background — and the treatment one could expect from a lighter or darker complexion.
Quadroon is derived from the Spanish word cuarterón, which is from cuarto, or one-fourth. So, roon isn't a separate word.
In the late-19th-century novel The Awakening, Kate Chopin equips the main protagonist with a quadroon nanny for her children. Notably, the servant has no name and makes no remarks throughout the entire book:
Mr. Pontellier's two children were there sturdy little fellows of four and five. A quadroon nurse followed them about with a faraway, meditative air.