I am a proud owner of a 2011 fit.

However, recently the fit has started to make a vibration noise (it isn't anything I can feel) at the back of it. I have checked everything, cleaned the car out, etc. to make sure it wasn't something inside the car making the noise. It only does it right after starting it up and will stop after a minute or two. What could be causing it?

 

Common problem the exhaust pipe metal shield "droops" a bit with age and starts being in contact with body parts. Then when it gets heated it moves enough for the noise to go away.

Go under the car and check the space around the shield.