Hm...it's actually a very interesting.
People seem to have forgotten that much of Q's involvement with Picard was essentially a long running philosophical argument over human nature, if men truly were good (or trying to be good) at thier core or if they were still driven by mere self interest, it's wasn't just Q going "Hey, I bet Picard woyld look really silly in running around in the woods dressed like Robin Hood". Though that debate that aspect was a bit more subdued in later seasons as the series moved away from the Roddenberry "we're more evolved, super moral TNG humans now" stuff into something slightly more plausible, it was still there up to the final episode. Picard eventually won that argument. Mostly.
Palpatine, by contrast, is humanity as Q portrays them to Picard. Petty, power hunger, self interested, self important, etc. But while Picard believed most humans were like himself, Palpatine is fully aware that most people are not like himself, so Q can't just do his normal thing of just testing someone's convictions and trying to prove them wrong.
Beat me to it. I agree, Q's style isn't to make droids do silly dances. Honestly, I don't see Q doing most of the stuff people are proposing here. That's not his style.
Q's method is generally to appear, dressed to give himself authority such as judges' robes or a grand marshall uniform, then make grandiose claims and put on trials. Sometimes they're literally just discussions of alleged sins, sometimes they're scenarios he invents where you have to overcome a personal flaw to win.
Part of the issue is that Sidious doesn't seem like the type to attract Q's attention. But let's go with it.
I'd see it as:
Q appears wearing Sith-themed armor identifying himself as a True Sith Emperor, employing elements of Marka Ragnos, Naga Sadow, Darth Bane himself, probably a few others too. He puts Sidious on trial for failure as a Sith.
What has Sidious done to deserve the lofty title of Sith Emperor? Clones to kill Jedi? That was Darth Plageuis' plan and Sidious just borrowed it. Killed his master? But not in a duel, was it? He was a coward who had to take the easy route instead of using his passion to gain strength, break his chains, and be freed by the force. Managed to corrupt Anakin? That was mostly the Jedi being idiots, we might as well give Sidious credit for Barriss Offee turning to the dark side. And what a pitiful apprentice at that, Sidious can't even stand the pressure of having a real apprentice so he had to settle for a mentally and physically broken wreck, after a chain of wrecks like Maul and Dooku who Sidious wasn't even able to fight and defeat himself. Pathetic.
Q then proceeds to create an elaborate scenario where Sidious has to actually prove his chops as a Sith. Perhaps, as he made Picard play Robin Hood Once, he recreates some event from Star Wars past/mythology and forces Sidious to actually handle difficult scenarios such as an apprentice far more powerful than himself that he can barely keep a lid on and needs to carefully finesse, ultimately Sidious can only actually win the trial by
losing and accepting with joy that he's raised an apprentice more powerful than himself and thus strengthened the Sith.
To me, I think Q might be more interested in challenging Maul and Savage when both were still alive. These are a pair who actually have room to grow and change compared to Sidious and Q could have them make a difference and change their lives, perhaps even coming up with a new philosophy.
