In the end we're talking about a character being given advice by a member of a species that's more often than not come across, if not as an antagonist, then certainly not as an ally. And the advice wasn't exactly delivered in the most earnest manner possible.
Would Picard have handled it any differently? I think we can already assume Sisko would have just punched Q in the face.
I think Lady Q may have nailed it when she suggested that on some level Janeway allowed her actions to be influenced by her brush with future Janeway in Endgame. That being said I doubt it was any level that Janeway was consciously aware of.
But we've also seen that, for better or worse, Janeway isn't the type to abide being told what to do. And this time around it bites her in the @$$.