I have defined an equivalence relation $\sim$ on a set $S$ and the quotient set $Q(S) := S/\sim$. Now I want to have a set $S'$ which only contains representatives of the individual equivalence classes, such that $\#Q(S) = \#S'$.
Is this definition correct:
$S' := \{ x | x \in [x]_{\sim}, [x]_{\sim} \in Q(S) \}$
No, that's not correct. The $S'$ you've defined is exactly the same as $S$. For example, suppose that $S = \{1, 2, 3, 4, 5, 6\}$ and $\sim$ is equivalence mod $2$. Then $Q(S) = \{\{1, 3, 5\}, \{2, 4, 6\}\}$. Note that, for example, $[1]_{\sim}$ and $[3]_{\sim}$ are both $\{1, 3, 5\}$ - so both are in $Q(S)$, and therefore both $1$ and $3$ are in $S'$. But that means you have two elements from the same equivalence class in $S'$.
In fact, no matter what $S$ and $\sim$ are, your definition of $S'$ will always be the same as $S$; every $x$ is always in $[x]_{\sim}$ by definition, and $[x]_{\sim}$ is always in $Q(S)$, by definition.
There's no convenient way to write $S'$ other than "let $S'$ be a subset of $S$ consisting of exactly one element of each equivalence class". Depending on context, you might be able to do better - for example, if $S$ is a finite set of whole numbers, you could use $S' = \{x | x \in S$ and $x$ is the least member of $[x]_{\sim}\}$.