When in Merchants' Den clicking on a player's name will display a list of options regarding that player. One of them is Send message iconSend Message.

While navigating on the World map World Map a click on any city-state will display related information on the left column. That info will also include the name of the ruler (player's name) and a Send message iconSend message link.

The Diplomacy adviser Diplomacy Adviser stores all messages (up to 14 days old). Both types of messages, sent and received, can be found there.