A "pet" dragon is not likely. I would even go so far as to say impossible, even for a WLDQ, but I'm not the final word on that. More likely, a dragon would keep a character as a pet...and by "pet" I mean "potential snack".
There is, however, precedence for some characters having some...favor with young dragons, but such a relationship can hardly be characterized as you suggest above. In such cases, the dragon remains fully in the control of GMs and is most likely not at a character's beck and call.
And for full disclosure, possessing a dragon egg will make a character into a target. Nevermind that the eggs themselves are fairly large and heavy. If a character somehow managed to sneak off with an egg, that egg's parent(s) would quite likely hunt it down, destroy the thief and bring the egg back to its lair.
That said, you're welcome to try! *GM grin*
