The first thing to try is to use constant functions (see Didier's comment). You will find that if you take $f$ and $g$ as arbitrary constant functions (e.g., $f \equiv 1, g \equiv 2$), all conditions hold.
Assume now that $f(n)$ is positive (i.e., $f(n) > 0$ for all $n \in \mathbb N$). Then by $f(n) = \Omega(f(n)g(n))$, you find $g(n) = O(1)$, and by further calculation, $f(n) = O(g^2(n))$ implies $f(n) = O(1)$. In fact, if you also assume that $g(n)$ is positive, all conditions hold iff $f(n), g(n) = O(1)$.
In other words, any two functions that are bounded and positive will do.
If one of the functions may have zeroes, more solutions may become possible.
Edit: As stated in the comments, "bounded and positive" should be read as "take values in some interval $[l,u]$ with $l,u$ positive".