The construction I think you are looking for is due to Benabou and given at the end of the paper "Some remarks on free monoids in a topos", LNM1488, Category theory (Como, 1990), 20–29. Here is a summary of how it goes.

Given a functor $U:A \to B$ with left adjoint $F$ one obtains for each $X \in A$ a simplicial object in $A$ with object of 0-simplices $FUX$, object of 1-simplices $FUFUX$ and face and degeneracy maps built from the multiplication and unit for the monad $T=UF$. If the adjunction is monadic and the monad is cartesian, meaning that $T$ preserves pullbacks and the naturality squares of the unit and multiplication are pullback squares then that simplicial object is in fact an internal category in $A$ with object of objects $FUX$. Let us call it $QX$.

For example consider the forgetful functor $U:Mon \to Set$ from the category of monoids to the category of sets. The free monoid monad is cartesian, so that associated to each monoid $X \in Mon$ you have a category internal to monoids, a strict monoidal category $QX$. If you take the terminal monoid $1$ then $FU1$ is the monoid of natural numbers $\mathbf{N}$ and you can check directly that the strict monoidal category $Q1$ is $\Delta$ with the usual ordinal sum.

Benabou does everything in the context you ask for - in a topos E with a natural numbers objects. He proves, for instance, that monoids in E are then monadic over E, with the monoid monad cartesian, so that you can associate a strict monoidal category to any monoid in E.

I don't think a natural numbers object in a topos need be of the form FUX for a monoid in E - I'm not sure as I don't know much topos theory. You would need the natural numbers object to be of this form for the above construction to give an internal category in Mon(E) with object of objects the natural numbers object.