Brian Monetti, Science is Awesome!
Quarks are a fundamental component of matter, and do not have smaller parts. Everyday matter is made of Up and Down quarks, which are the two most common of the six types of quarks. The other types, called flavors, are Top, Bottom, Charm, and Strange, and are only produced in extremely high energy particle collisions. This happens in particle accelerators, in cosmic rays collisions, and possibly inside very dense stars, but not in everyday matter that you would encounter in your life.