Let $(\mathbb{N},\ 0\colon \mathbf{1} \mapsto \mathbb{N},\ s\colon \mathbb{N} \mapsto \mathbb{N})$ be a natural numbers object in a topos $\mathbf{T}$. Elements of $\mathbb{N}$ are maps $n\colon \mathbf{1} \mapsto \mathbb{N}$. Note we can generate an infinite family of elements of $\mathbb{N}$:
$$ 0\colon \mathbf{1} \mapsto \mathbb{N} \\ 1 = s0\colon \mathbf{1} \mapsto \mathbb{N} \\ 2 = s1\colon \mathbf{1} \mapsto \mathbb{N} \\ \vdots $$
Intuitively, this should generate all the elements of $\mathbb{N}$. Is it true that every element of $\mathbb{N}$ is in the above family? If not, can we put conditions on $\mathbf{T}$ to ensure this?
No. For instance, in $\mathtt{Set}\times\mathtt{Set}$, the natural numbers object is $(\mathbb{N},\mathbb{N})$ and its elements are given by pairs $(a,b)$ of natural numbers, with $s^n0=(n,n)$ so your family of elements is only the "diagonal" elements.
I don't know what sort of condition you are looking for that would guarantee this. In a Grothendieck topos, this is equivalent to the topos being connected. But weirder things can happen in elementary toposes; for instance, if you take an elementary topos that comes from an $\omega$-nonstandard model of set theory, it will be connected but the elements of its natural numbers object will be a nonstandard model of arithmetic.