I would also picture Sherlock and Q getting on better than Sherlock and Mycroft, being closer in age, while Mycroft took on a more parental role that tended to make him an authority figure to rebel against.

I can totally see that. Also for whatever reason, both Sherlock and Q have a similar sense of humor. At least to me. So I can see them getting on in that regard. At least when they're not trying to one up each other with Sherlock deducing embarrassing things and Q hijacking Sherlock's website.

John: I'm not entirely comfortable with the idea that my banking information is at the mercy of someone who's related to you, Sherlock.

Sherlock: You should be more concerned that with one keystroke he can level the value of the British pound.

John: Yeah...but...he wouldn't?

Sherlock: Did I never tell you the story behind why the euro was invented?