If you are on Visit visa status and have been over staying in Qatar problems are your passport might have been given to the CID by your agency ex. Fahad,cleopatra and evryday of lapse visa will be equivalent to a certain fine. They will ask you to pay a certain amount to get your passport so you can get back to your home country.
If your question is arresting you in some criminal offense like carrying dangerous alcohol, drugs etc.
then yes you might be deported.In some cases your relatives or friends in Doha will not even know you have been deported.