No, he wouldn't. Should he have put Voyager back on earth, it wouldn't have been Janeways choice to grant / reject Quinn asylum. It would've been up to the federation. And with Picards experiences with Q (de Lancie), they would probably not have been too fond to have a member of the Q continuum in their midst.
"He's devious and amoral and unreliable and irresponsible and… and definitely not to be trusted."
– Jean-Luc Picard, TNG S4E20
And as we know from TNG:Attached (S7E08):
"We still plan to take our petition directly to the Federation Council! They'll listen-!"
"They will also listen to the reports of the Captain of the Enterprise and his First Officer! And I can tell you right now the First Officer's report will go something like this: 'Kesprytt, a deeply troubled world with social, political, and military problems that they have yet to resolve. The Kes, while a friendly and democratic people, are driven by suspicion, deviousness, and paranoia. It is the opinion of this officer that they are not ready for membership.'"
they'll probably ask Picard ,as one of the only ones having had more than one contact with a Q, of his opinion, who's not really fond of Q.
That they are not free of prejudice is shown every time they meet Q. Every time Q appears, Picard is less than thrilled. If humans were so evolved as they think they are, Picard wouldn't hold such a grudge against Q. Au contraire, he made Starfleet aware of the Borg. So, simply being inconvenient and having some jokes at Picards cost was enough to make him unwelcome. That's perfectly normal but also shows that 24th century humans haven't really overcome the flaw of prejudice. And a Q might be more than just a little inconvenient for the federation but a huge risk.