I encountered the title question answering this question. It is well-known (see, for instance [vdW, $\S$ 43]) when $q$ is a power of a prime number there exists a finite field of order $q$. In this case a Steiner System $S(2, q, q^2)$ can be realized as a finite affine plane. I guess that the answer for general case may be already known, so I googled for it, but failed to find it.
References
[vdW] B. L. van der Waerden, Algebra (Russian edition).
This is a notorious unsolved problem in combinatorics: are there any $n$ other than prime powers for which an $S(2,n,n^2)$ (or equivalently an $S(2,n+1,n^2+n+1)$) exists? The Bruck-Ryser-Chowla theorem implies that if $n\equiv1$ or $2\pmod 4$ then if an $S(2,n,n^2)$ exists then $n$ is a sum of two squares. This eliminates $n=6$.
The $n=10$ case was eliminated by Clement Lam, using a vast computer search in c. 1989. Apart from these, little is known.
