Possibly, but we don't see them in the show. The only mention of any race of alien that "abducts" humans is in the Voyager Episode "The 37s", where we learn that a race in the Delta Quadrant, the Briori, frequently kidnapped humans to be kept as slaves. However, the human's revolted and drove the Briori off long ago, and we never get a chance to see what one might look like.

Possibly related to the question is the Deep Space 9 episode, "Little Green Men", where we learn that Quark, Rom, and Nog get sucked back in time and are inadvertently, responsible for the Roswell incident.

Interdimensional aliens that abduct crew members from the Enterprise do exist. They are seen in the episode "Schisms". It is not established that these are in fact the same grey aliens reported in earth's history, but their behavior of abduction, experimentation, and return are similar.

Schisms (Episode)

However, they do not physically resemble descriptions of greys.

I know this alien was a ruse, but there must be some veracity to it, since Kirk and crew accept it so readily. It doesn't seem reasonable that Balok would have chosen a completely fantasy disguise to try and fool rational adults, I would assume these dudes exist in-universe.

(He is an avatar used by a character Balok in the episode "The Corbomite Maneuver", and also appears in the end credit sequence of every episode)

We know that the aliens recovered from the crash at Roswell, New Mexico in 1947 were ferengi; as this is one of the roots of the grey alien myth (along with Betty and Barney Hill), I'm inclined to say that grey aliens are generally other species that have been (deliberately?) misidentified.