I personally would not - the Jones Falls (which feeds into the harbor) historically has had an issue with sewage runoff. There is also a ton of industrial pollutants in the sediment, including stuff from a chromium plant.

If you don't mind driving a bit there's a ton of good spots in the area - a lot of people fish right below the Conowingo Dam (go when they are running water through the turbines), also Sandy Point State Park. MD DNR also does a lot of trout stocking, I'm at work so I can't provide specific locations, but if you look on their website you should be able to find it. Lastly I would say go and get a fishing licence, I would imagine any body of water you need a permit, it's just that enforcement is spotty.