What symbol gives the count of elements in a set?

If there is no misunderstanding from context then you can use $|A|$ to denote the cardinality of a set $A$.

However the symbols $|A|$ can be interpreted in many different ways (by example as the determinant or the norm of a matrix $A$, as a Lebesgue measure of the set $A$ or some absolute value). If you want to avoid confusion you can use the symbols $\# A$ or $\operatorname{card}(A)$ instead.

In any case when you want to use some symbology it is recommended to define it meaning prior to it use. Then you can use any symbolization that you like or want once you had previously defined it meaning.

The notation for set cardinality (number of elements in a set) is $$|A|$$