Is there a trolley in the city close to a parking lot?

Yes, but we prefer the term street car. The nearest stop is actually a little ways over the staircase on the opposite side from the cathedral.

Walking from bourbon street to Jackson square

Yes

Didn't see a trolley ,but there is a lot of city parking lots be within walking distance to Jackson Square and french quarter. New Orleans is popular on mule drawn carridge tours that start right by Jackson Square

Yes, in the city you have a lot of options to park the car