A first contact specialist was an individual with advanced training in making first contact with alien species.
Commander Saru was a first contact specialist, and as such took the lead when his away team encountered the Pahvans on stardate 1308.9. (DIS: "Si Vis Pacem, Para Bellum")
Tam Elbrun was considered the Federation's "finest specialist in communication with unknown lifeforms". This led to Starfleet assigning him to the USS Enterprise's 2366 mission to investigate the mysterious "Tin Man". (TNG: "Tin Man")