If your system administrator has process accounting enabled (the way to do this varies from system to system), and the script is invoked by name (as in ./script or $HOME/bin/script); not by invoking a shell and giving the script as an operand (as in sh script, or bash ./script, or ksh -xv $HOME/bin/script) then you could search the process accounting database to see how many times the script was run, when it was run, and by whom.

If the script itself updates a log file that logs each time the script is run, you can search its log file.

Otherwise, you're probably not going to find what you want.