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.

Oh hey, another one of these:

<Multi_key> <Multi_key> <c> <o> <l> <l> <e> <c> <t> : "" UE03D # FFXIV Collectible mark

If you want to show how collectable you and/or your friends are.

(I also went back and tweaked the old one. I didn't realise at the time that it's a standard thing to include the codepoint in the Compose definition.)