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-D's 2366 mission to investigate the mysterious "Tin Man". (TNG: "Tin Man")