Depends on your budget.  There are some adventure tourism companies in Doha which organize fishing trips.  Could be either a speed boat or a traditional dhow.  Sometimes the results are quite disappointing, since these boats have to ply a pre-determined route, which has been laid down by the coast guard.  The beaten path has been overfished.  Night fishing by expats in boats was previously not allowed.  You can check current status now.  The weather is cooling down a bit.  Maybe you could wait a bit longer, or you might get cooked on the boat in the sun.

Better option and more fun would be if a Qatari in your office has a boat and you go with him.  Then I guess night fishing would also be possible and you can go off almost anywhere, where there are plenty of fish.

If you would like just to test the waters initially, go down to the Corniche on a Friday morning and you would see a large number of people fishing.

Have you ever tried kite fishing?