^The Children Of Tama
Nope, I don't think they count.
"First contact" had previously taken place; Enterprise's mission was to attempt to establish communications.
"Several" previous attempts had failed, so first contact had already occurred.
The Memory Alpha article only notes three examples, Q, Ferengi and Borg, fyi.
But does it count as first contact if no meaningful communication takes place? If all that's required is for both species to be aware of each other's existence, doesn't that set the bar really low?
As I recall, the Federation and the Ferengi were very aware of each other, but it took Picard to officially make first contact.