Tested with quassel (0.12.4) and qutebrowser (1.0.4)

Inserting emojis using xdotool is not possible:

xdotool type --window <id> "emoji"

(I could not insert a real emoji in the example because of the following error: "We can't create this issue for you right now, it could be due to unsupported content you've entered into one or more of the issue fields.")

Copy/pasting the character from both primary and secondary selection/clipboard works fine though.