Notation for expressing function call count during a program run
Based on your desired usage and the fact that there likely isn't a widely-known notation for this, I think that, whatever notation your use, you should formally simply consider it a name. As you want to use the actual value, a variable (or function if you wish to distinguish among parameters for the call) would suffice.
What I mean by considering the notation a name is that you give your notation no syntactic status: it is simply a naming convention for some function or variable. This requires that you must define the name for each different method call number you wish to describe. I advise you to do this, as number of function calls is something that is context-dependent (calls by whom? calls made when?), so defining this for every function call separately gives a natural place to specify this.

So, for your particular example, a name like $\#\mathrm{function1}(\bullet)$ could work.
Then, we can write something like

Let $\#\mathrm{itersearch}(a)$ be the number of calls to the function $\mathrm{itersearch}()$ in the execution of algorithm A with parameter $a$.
(...)
Most of the time less than $\#\mathrm{iterSearch}(searchState)/3$ child nodes were searched during experiments.