I figured out how to have WeeChat in Termux create a notification on your android device whenever someone mentions your username or you receive a private message.
You need to have Termux:API installed for this. All you need to do is type following command into WeeChat:
/set trigger.trigger.beep.command "/print -beep;/exec -bg termux-notification -t 'IRC Notification' -c "${tg_tag_nick}: ${tg_message_nocolor}";/exec -bg termux-vibrate"Explanation
Normal WeeChat behavior is to send a BEL character to the terminal whenever someone mentions you or sends you a private message. This extends that with the commands termux-notification and termux-vibrate found in the Termux:API. The content of the notification is ${tg_tag_nick} (which is the user who sent you the message) and ${tg_message_nocolor} (which is the message without coloring). Here you can find more available variables.