I just bought a GE Adora gas stove with convection oven. I wanted to know if it was normal for the convection fan to start and stop about every 5 seconds during the cooking cycle

I don't have a GE but I would say no. Mine doesn't ever stop.

Mine does not stop either.

You have a winding that is open with a tiny gap and as it runs it opens more and turns off the motor. This open winding could be in the motor or in the relay that controls the motor. It is hard to find, but with a good digital meter you should be able to find the problem.