Hello there,
generally landlords will not rent accommodation for 1 or 2 months. There are a few exceptions:
- some serviced apartment buildings offer this. You'll find most in the Westbay/Dafna area
- some hotels have apartments or suites with more than 1 bedroom and also offer monthly rates. You'll find quite a few in the Corniche area close to the Moevenpick Hotel (old one) and in Westbay/Dafna
- be prepared to pay a premium as this is considered by many businesses a "special" condition

For a list of hotels in Doha consult the Qatar Tourism Authority website (google it).

Regards.
(www.ahbqatar.com)