We've visited Quetico in May through September and have had both good and bad experiences weather wise any time .  May can be great, but it's hitting the right slot after spring storms/rain and before the bugs hatch.  June can be pretty buggy.  July and August seem (for us at least) to have the most reliably good weather, though in a week trip you can count on at least a day or two of high wind and/or thunderstorms.  September seems to be anything from soup to nuts -- I've had a week of glorious sunshine with highs in the 70's, lows in the 40's and I've had a week with 45 degrees and rain every day....  Guess there's no right answer...........