Aspire Park and the park along the corniche have free wifi but I wouldn't recommend going on Fridays.

Many of the coffee shops have wifi as well and they are probably quieter than some of the parks.

Whether you would be allowed in the university libraries at Education city is another question. There are security checkpoints at all points of entry and it is highly doubtful you'd get past the gates at Qatar University.

You may be pleasantly surprised at some of the coffee shops and certainly using your computer at the parks (when the weather improves).