On Unix, no - waiting on external services (API calls, databases, etc) does not count towards the max execution time. On Windows, the time refers to the actual wall-clock time.
From the manual:
The set_time_limit() function and the configuration directive max_execution_time only affect the execution time of the script itself. Any time spent on activity that happens outside the execution of the script such as system calls using system(), stream operations, database queries, etc. is not included when determining the maximum time that the script has been running. This is not true on Windows where the measured time is real.