Some context if it helps: I'm learning Bayesian filtering and there's a bunch of notation I don't get so I'll probably ask a bunch more questions including this one.
Small excerpt from the page:
changes in the environment take place at discrete, equidistant time steps $t \in \mathbb{N}_0$, where sensor measurements arrive at every time step $t \geq 1$.
I have looked at this question and this question and I've concluded that it most likely means all natural numbers from 0 upwards, since anything mod 0 is basically infinity.
Short version: it is almost certain that $$ \mathbb{N}_0 = \{ 0, 1, 2, 3, 4, 5, 6, \dotsb \}. $$
In more detail: the question about the integers modulo $n$ has nothing to do with the notation $\mathbb{N}_0$. The notation $\mathbb{Z}_n$ generally refers to the quotient group $\mathbb{Z}/n\mathbb{Z}$, which is the group of integers modulo $n$. That being said, it is worth noting that if we assume that $\mathbb{Z}_0$ refers to the quotient group modulo zero, then $$\mathbb{Z}_0 = \mathbb{Z}/0\mathbb{Z} \cong \mathbb{Z},$$ since $a-b = 0 \pmod{0}$ if and only if $a = b$. Thus "division by zero" (in this sense) is actually okay.
The notation $\mathbb{Z}_p$ can also refer to the $p$-adic integers, where $p$ is your favorite prime number, but that, also, is neither here nor there.
The other question that you cite is a bit more to the point, though still not quite right. Yes, $\mathbb{Z}_+$ typically refers to the positive integers, and $\mathbb{R}_+$ is the positive real numbers. However, this is a notation that specifically deals with ordered groups, rings, or fields. In such a setting, we want to have a notation for the positive and negative elements, so there it is. In the case of $\mathbb{N}$, everything is positive and $\mathbb{N}$ isn't even a group, anyway (though it is a monoid!).
In the context of your question, it is nigh certain that $\mathbb{N}_{0}$ refers to the set of nonnegative integers. This is the set of all natural numbers, along with zero (or, if you believe that $0$ is a natural number, then $\mathbb{N}_0$ is the natural numbers, with the notation indicating that you believe that $0$ is a natural number, and you don't want anyone to be confused). In other words $$ \mathbb{N}_0 = \{ 0, 1, 2, 3, 4, 5, 6, \dotsb \}. $$