Comment by Gro-Tsen on Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$ in a topos?

A formal answer to your question has already been given by Simon Henry. But let me make the following more informal remark: even just classically, the whole point of modal operators is that they give rise to a semantic of “possible worlds” (Kripke semantics) where the truth value differs from one world to another: so the whole point of $\Box$ is that the truth value of $\Box p$ doesn't just depend on the truth value of $p$. As such, it is unreasonable to try to make $\Box$ into a function from the object $\Omega$ of truth value to itself.