Nah, the Hollywood aliens came first, and within months of grays showing up on TV, random people telling their stories started matching the Hollywood descriptions.
The first ever mention of beings that looked like greys was in an H.G. Wells story. His story wasn't actually about aliens, but about evolved humans in the future.
The first reference to greys outside of H.G. Wells was in a Swedish 1933 sci-fi story:
The first depiction of greys on television were on the Twilight Zone episode "Hocus Pocus and Frisby", which aired on 13 April 1962.
The second depiction of greys on televsion was on The Outer Limits episode "The Bellero Shield", which aired on 10 February 1964.
The first ever mention of "grey aliens" like that in real life is the Betty and Barney Hill story. Betty and Barney Hill recovered their experience during hypnosis...in late February 1964, just two weeks after that "Outer Limits" episode aired. Their actual "abduction" had come three years earlier, in 1961, but they NEVER described the aliens as grays until after the Outer Limits episode.
The second mention of greys in real life is the Roswell incident. But there are ZERO recorded descriptions of the Roswell incident mentioning greys before the 1960s. All of the witness testimonies that describe the aliens as looking like grays came well after the Betty and Barney Hill story was publicized.
The fact that gray aliens with big bald heads and wrap-around eyes began showing up in abduction stories immediately AFTER silly shows like The Twilight Zone and The Outer Limits depicted it seems like too big a coincidence.