I called the city the first year I moved here (2006) to inquire about what was needed to put in a patio cover. Laurie is correct, as I checked the HOA and as long as the city is fine with it, the HOA is ok with it too. From what I understand getting a permit is fairly easy, and should be done before any work is performed. In my case, IIRC (if I recall correctly), I had to provide a drawing of what I wanted to do, a patio extension is much simpler, so I don't think it would be a big deal.
Definitely call the city and ask them, I would also end the call with asking "I'm new to this, is there anything else I should be aware of or might need to know?", they may mention. I believe it would need to be inspected and then signed off on.
A neighbor catty-cornered across from me put in a sidewalk and I remember seeing an inspector out there a couple of times, just before and after their front walk was put in (going to their front door), but it seemed to only take a day or two for the whole process.
Good luck to you!