After we flush the toilet in the master bath it will let out this sound that is like a low flying commercial airplane, and it last probably 2-3 minutes. It gets louder before it slowly dwindles off. You can hear it clear across the entire house (one level home). I don't have the slightest idea what it could be, and calling a plumber sounds stupid, right now. I want to have some idea of what I am dealing with, before I involve a plumber and a bill.

You have a vibrating washer. That washer could be in the fill valve of the toilet or the stop valve under the tank.
Which type of fill valve do you have?