When I run my shower it will sometimes begin whining loadly, a high pitched kind of droning thing. I can get it to stop if I turn off the shower head (switch to tub) and
Then back on to shower. Then, it will come back later
A few minutes.

I am on a well.

A plumber once told me that this is due to tiny gravel that gets caught in the shower head. Is that true ?

The first place I would check would be the shower head. Remove any trash you may find and if there is a black neoprene washer with small holes around the outside pry it out and throw it away. This is a water saving device that sometimes squeals. You don't need it to make the shower head work and it will give you better pressure. If after all that it still makes noise I would check the diverter. Some diverters are on the tub spout and some are a part of the valve itself. A good way to check the diverter is to turn on the shower with the shower head removed. If it's quiet then the diverter's OK, if not then you have located the source of the whine. Hope this helps and thank you for rating my answer. Tom