As c256 surmised, I use kill to send signals other than SIGKILL. I don't think I've ever even known about the C-\ or C-y bindings for SIGQUIT. I also hardly ever use shell job control, to the extent that I have an alias fg that does something unrelated, so I would need to say \fg to get the shell foreground command.

But if for some reason I wanted to do that, I would use C-\ : send-prefix. That sends the prefix character to the process running in tmux, in this case the shell. It's also super useful if you ever find yourself running a nested tmux (i.e. in an ssh session) with the same prefix character. You can just send-prefix d to disconnect the remote shell from the remote tmux session, then open a new local shell outside of your local tmux session, log in and reconnect to the remote tmux.