I am running a homebrew game and I am currently working on designing a Temple/Possible group patron (TCoE), and the Zhentarim's way of communication with their members interested me. Is there any cool way you can come up with for a Couatl Temple to send messages that might be similar to the Zhentarim's? (i.e. Trained messenger "pigeon")

Given their general and quite powerful spell lists, I wouldn't hesitate for a moment for Couatls to be able to use message or even sending to communicate with each other. Sure it's not in the default list of their spells, but you could either add or swap out for your setting. 

I'd also vote for sending. Instantaneous. Can't be intercepted. Can't be read by anyone else.  It's pretty perfect for organizations with spellcasters. Or a tomelock with farscribe invocation, if they want to not use spell slots.

Right in the stat block, I'm assuming Couatl are trusting of their represenatives or people otherwise charged with a duty to the Couatl and not very micromanagerial, so the once a day dream is probably the best way for a Couatl to be sure the recipient gets the message.