I cannot get either Farkas or Vilkas to do the Purity quest to cure their lycanthropy. I have done over 40 of their radiant quests in a row and neither of them will offer this quest. I have cured myself of lycanthropy and have done the Totems of Hircine quest with Aela. Otherwise, I am not sure what else could be affecting this quest. Am I simply having horrible luck with it not activating, or am I doing something wrong?

i think its random

This has happened to me, so I'll speak from experience. This usually occurs if you have the hag heads in your inventory and then go do Diplomatic Immunity. That quest removes the heads and once it's over, places them back in your inventory, oftentimes with the problems you described. If that is the case, there's honestly nothing that can be done at this point. If you have the UP's, you can try dropping the heads on the ground, then speak to either brother, and then pick the heads back up again. That can sometime fix the issue, though not always.

See at the Bugs section: http://www.uesp.net/wiki/Skyrim:Purity