I am looking for the explicit reference to the fact that for a quasi-projective scheme a class in the étale cohomology of a sheaf of a certain degree can by computed using Cech cocycles.
This result is due to Artin and can be found in the paper "On the Joins of Hensel Rings", Corollary 4.2. There the assumption on X is that it is Noetherian and every finite set of points is contained in an affine. If X is quasi-projective over a Noetherian ring, then this is true (as Artin points out in the paper).
There is no good definition of a quasi-projective scheme without reference to a base scheme. Maybe you mean: a quasi-compact and separated scheme X which has an ample invertible sheaf? In this case Artin's result implies it for X by standard limit techniques.