Would there be any genetic diversity within the population of drones from
the same queen? I don't think so but intuition nudges me to think about
this.

Nope, because they only get DNA from the mother (no father involved), therefore no recombination. This is no doubt why bees fly far off to mate, to minimize the chance that the drones a queen mates with will be her brothers, but identical twins as well. In the real world, she will mate with a mix from many different colonies, with which she is not directly related. 

On the other hand, the bees from a given geographical region will constitute a population and together they may evolve in a certain direction, eventually adapting to that region, and over millions of years of separation, become genetically incompatible with other populations, like what probably happened with A. mellifera and A. cerana.

PLB

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