QR
Initial applications are "screened" by the Human Resources Department, prior to being filtered (if appropriate) to the department heads who are requiring the posts to be filled.
The head of HR is a Mr. Asadullah. However, I suggest that you do NOT call him as he is extremely busy and he will probably not welcome the intrusion. Besides, your call will not likely receive the attention it deserves.
My recommendation, which is not all that ideal either, is that if you know someone in Qatar Airways or someone who lives in Doha already, is to ask them to pop in briefly to the office on your behalf and ask to speak to one of the HR assistants who can at least tell you on whos table the application is currently lying at this time.
Sorry can't be of more help. but the offices are currently swarming with people and applications......