The problem of finding a formula for the clique number of the Paley graph G*(p) where p is a prime number such that p ≡ 1(mod 4) seems to be a hard one. These clique numbers are known for the caws 5 ⋚ p ⋚ 1601. In this paper we show that the clique number and chromatic number of the Paley graph G*(p2r) are both pr, where p is an odd prime and r is a natural number. Upper and lower bounds for the clique numbers and chromatic numbers of some related graphs are also determined.