How I guess it will go:
1) Q will make Picard question his very existence - after all, Picard did technically die and now lives on as an android. Wandavision touched on Ship of Theseus, and while another show already went there, given Picard's background it would be a hard thing to ignore in this show. Is he Picard, or just a copy, or neither?
2) He will also call on Picard to question is views on mortality - this version of Picard has similar limitations/life span, but could easily extend that. I suspect Data's decision will also weigh into this
3) Q will also focus on Picard's current situation due to his age. Ties heavily into the 2nd one.
Q usually is at his best when he is trying to "teach a lesson" and trying to test Picard/push Picard to better himself. If they stick to that version of Q, then they could (hopefully) make something good.