You got that the wrong way around. Hadrons are made up of quarks. There are different types of hadrons, e.g. baryons (3 quarks) and mesons (1 quark, 1 antiquark). So a proton is just one special type of baryon and therefore also a hadron.
Now what quarks are made up of is not a sensible question in the frame we use to think about particle physics nowadays. Quarks are the smallest units or building blocks of the Standard Model of particle physics and therefore cannot be split up into further constituents.
On the other hand, if one chooses to believe in string theory, basically everything is made up of tiny vibrating strings, including quarks.