No, I do not.
MacGuffin-as-Character. When the writers needed to stir the pot they could cart him in as a daemon ex machina.
Q's Human Moments Were Redundant. A favorite story is that of the alien who must grasp with what it means to be human. But Trek didn't need yet another turn of this wheel since it has always had characters like Spock and Data as members of the crew. Listening to Q gripe about needing to eat makes for light comedy, but sheds no light on the human condition.
The Message of Q's Character Was Redundant. We learned from Gary Mitchell and Trelane (and how many other characters in TOS?) that power corrupts. We didn't need a slightly older version of Trelane.
Ridiculously Overpowered Characters are Not Interesting. Superman suffers from this problem and has occasionally had his powers scaled down from time to time to keep things interesting. Q, however, is more powerful than most comic book characters.
The Character Does Not Project Evolution of a Species. Q is not more intelligent than humans. He claims a dubiously high IQ, but gets outwitted by Picard on a regular basis, so he's no super-genius. In terms of EQ, Q is far behind the humans he torments. In actuality, the ability to process the amount of data he claimed to be able to process - and by process I mean directly cognize it - he directly apprehended his knowledge (unlike humans who grind out correct answers from calculators) of the universe. A creature like this would see patterns (historical, emotional, moral, mathematical) that no mere human could ever hope to see. Such a creature would have such a sublimity and nuance in thought as a result. And yet the writers imagine that such a power of intellect would only mean (1.) You would get a spoiled brat, (2.) and or a power hungry egomaniac looking to rule the whole universe. Yes, the next big step in evolution on Star Trek is always a big step back, because - as always - power simply corrupts. And Q is absolute power, so how could he be anything but corrupt?