As far as I know you can only use a foreign license for a maximum of 1 week then you must get a temporary license or apply for a full one if you have an RP. When you apply for your Qatari license you will not be fined for not applying earlier because you're not really going to tell anyone that you've been driving illegally for months, are you??!!
I'm pretty sure that a Euro license can just be swapped for a Qatar license without a test, just need to do an eye test. Good luck!