You can just work out x^2 mod p for x=1,2,3,4,5....(p-1)/2.
They will be distinct.
It may help to know that the set of all quadratic residues is a subgroup of index of the multiplicative group of the integers I've been thinking about this recently and trying to develop a group-theoretic approach to results about quadratic residues; I'll keep you informed about further progress I make.