Can anyone recommend a stretch of beach/a resort in the Port Douglas/Cairns vicinity?

Turtle Cove would be my recommendation, in a beach front room. Guaranteed no children. Almost all guests are gay men. 

Another option, but south of Cairnsand close to Mission Beach, Bedarra Island is luxurious and not really a family resort (unlike most in the Cairns/Port Douglas area).

I can strongly recommend the QT Resort. Calm and spacious with a nice deign chic, but still with a calming subtlty to it.