Is there a quick way to write say positive integers in an interval in mathematical notation?

For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,

Is this the quickest way?

$x\in \left[ 1,50\right] \cap \mathbb{N} $

Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$, $\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.

It depends on your own preference on how to write things down, there are countless variations, for example

$x \in \{ n \in \mathbb N : 1 ≤ n ≤ 50\}$

$x \in \{1,2,...,50\}$

$x \in \mathbb N_1^{50}$

A common convention in French is

$$ x∈⟦1, 50⟧ $$

and I am genuinely surprised to learn that it might not be common elsewhere ! In any case, $\{1, …, 50\}$ or maybe $\{1, 2, …, 50\}$ should be universal and more readable for most people.

For your other question, still from the French perspective, $$ \mathbb{N} = \{0, 1, …\}\\ \mathbb{N^*} = \{1, 2, …\}\\ $$ though the second one is sometimes frowned upon due to it being an abuse of the $A^*$ notation (where $A$ is a ring) that leads to confusion for the $\mathbb{Z}^*=\{-1, 1\}$ case.

I have never seen $\mathbb{Z}^+$ used, but if I had, I would probably have assumed $\mathbb{Z}^+=\mathbb{N}$, following $\mathbb{R}^+=\{x∈\mathbb{R}|x⩾0\}$.

For the specific case that you start at $1$, it is fairly standard in combinatorics to write $[n]$ for $\{1,\ldots,n\}$, so $x\in[50]$ would work. This doesn't really help for other ranges, though - you could write $x\in[50]\setminus[10]$, but you probably shouldn't :)

To answer your other question, I prefer $\mathbb N$ to be $\{0,1,\ldots\}$ and $\mathbb Z_+$ to be $\{1,2,\ldots\}$, but there is no consensus on the first, and it's probably safer to write $\mathbb N_0$, which is unambiguous. Even $\mathbb Z_+$ could be misinterpreted, but I think when writing in English it's standard that this does not include $0$ (when writing in French, I'd expect the standard to be different, but I have no first-hand knowledge of this).

I do wonder why so many people believe convoluted notation is better than plainly writing what you mean.

"Let $x \in \mathbb{N}$ with $1 \leq x \leq 50$."

The twin purposes of notation are clarity and precision. Use of new or rare notation subverts both. Excessive density subverts clarity. Use of a single natural language word for exactly its meaning is both clear and precise.

Anyone will understand

$$n\in\{1,2,\dots50\}$$ or even

$$n\in\{1,\dots50\}$$ without toil.

If it is clear from context that $n$ is an integer,

$$n\in[1,50]$$ is good enough (and is very compact from the standpoint of LaTeX formatting :) ).

Following @EspeciallyLime, $[50]$ is a good option, though you should introduce the notation. This remains compatible with more general intervals like $[11,50]$.