You can state that quicksort has run time O(n^2). The function you're describing with O(n^2) is f : S -> R, where S is the set of input values and R is the set of real numbers, and f(x) is the running time of running quicksort on input value x, and n : S -> N is the size of input value x. The notation O(g(n)) describes the function f in terms of the function n.
That f is in O(g(n)) means there exist constants C and N_0 such that for all x in S, provided that n(x) >= N_0, it is true that |f(x)| <= |C g(n(x))|.
And that f is in Omega(g(n)) means there exist C > 0 and N_0 such that for all x in S, provided that n(x) >= N_0, it is true that |f(x)| >= |C g(n(x))|.
That f is in O(g(n)) means there exist constants C and N_0 such that for all x in S, provided that n(x) >= N_0, it is true that |f(x)| <= |C g(n(x))|.
And that f is in Omega(g(n)) means there exist C > 0 and N_0 such that for all x in S, provided that n(x) >= N_0, it is true that |f(x)| >= |C g(n(x))|.