$F(x,y)$ is the statement that "$x$ is $y$'s father." There are two domains for the statement $$\forall x \forall y \exists z(F(x,z) \land F(y,z))$$
such that the statement can be true or false, depending on the domain.
I interpreted the problem to read in English as $x$ is the father of $z$ and $y$ is the father of $z$. This would imply that $z$ has two fathers, $x$ and $y$. In terms of coming up with a domain, this question seemed peculiar in that I wasn't sure whether $x$ or $y$ would be sets of one or more people.
If I were to say that the domain of $x$ included all fathers and the domain of $y$ included all stepfathers, the statement might make biological sense but not logical sense because the statement reads "$x$ must be $y$'s father."
Am I interpreting the question correctly or overthinking it?
On just about any nonempty domain of people who are or have been alive the statement is false: there will be some member $x$ of the set who is not the father of anybody in the set.
On the empty set the statement is true.
If you allow stepfathers as fathers, a nonempty domain for which the statement is true is $\{Oedipus\}$: by marrying his mother, he became his own stepfather.