For the last 200 years or so the monarch's official birthday has been marked by Trooping The Colour - this is an open air ceremony so the official birthday is always in the summer. I believe that other countires have similar customs. So really the date has no significance other than it being a day set aside for pomp and ceremony when it hopefully won't rain!
The Queen's official birthday (UK) is the day on which she was crowned, ie 4th June. This is marked by the Trooping of the Colour on the nearest Saturday but the fact that it is in the summer is pure coincidence.