I am assuming you and your partner are not Muslims. Being married has its advantages in terms of benefits an employee gets from the employer; housing allownces are higher, type of accommodation provided. Giving birth at a hospital here requires proof of marriage. Hotels won't give you a room unless you're married.
A Muslim marriage is the actual marriage contract conducted by a Muslim cleric, Sheikh, for a Muslim bride and groom. Hence to do that you would have to become Muslim. I don't think it's common for Westerners to consider this option.
Having said that I know some couples co-habit and they haven't run into problems.
If you don't intend to become a Muslim out of conviction why not tie the know where you are? Or, I suppose, you can get married at one of the churches here.
Also don't show too much affection in public. Holding hands is ok but no kissing or-over-the-top hugging; a quick "hi" or "bye" hug is alright. NO KISSING IN PUBLIC, not in a mall, a taxi, a restaurant...
Other than that Qatar is a nice place to live. My family likes it and the locals are the most decent I've met in the Gulf.