Kewlio: I'll see you guys next time

Here's a compose string for your .Xcompose file:
<Multi_key> <h> <q> : "" UE03C # FFXIV High Quality mark

That's Unicode codepoint U+E03C, which is in the private use area. SE uses it in FFXIV for the High Quality mark. So, all you have to do is type [Compose]+h+q and voilà.

I haven't tried it in Linux (mostly because I still struggle to set up a copy of Linux for gaming/streaming), but it works with WinCompose.