At random times I get a fog horn noise that builds in volume. Sometimes it just goes away and some times I feel a vibration throughout the bathroom. Flushing the toilet seems to help but this noise is also happening when I sit on the toilet. There is no water moving when this sound happens. What is it?

Most likely trash or debris in the toilet fill valve.
Which type of fill valve do you have?

I think Harold is on the right track. If this is the issue you have 2 problems. The fill valve washer is vibrating AND your flush valve is allowing a small amount of water out of the tank into the toilet. That would periodically caused you toilet to refill even if nobody flushed it.