The Barzan originally showed up in TNG 3×08: The Price. There it was Premier Bhavani heading negotiations for the Barzan Wormhole. They also make in appearance in DIS in the form of Commander Nhan.
Outside of these two characters, however, little is known about the Barzan. I worked with Callisto Wojcikowski, a Barzan enthusiast from the Clear Skies Discord, to come up with the species profile you see below.