When watching Ultraman Gaia with my kids as it aired for the first time and seeing Gan Q appear in the show, the first thing the monster reminded me of was Ganmons.
Gan is the Japanese word for eye while the word me (/meh/) is more customarily used, and, as gankyu means eyeball, Gan Q is a sort of phonetic equivalent or pun of the term.
While I don’t know much about Gan Q featured in the recent series after Ultraman Gaia, I think the Gan Q that appeared in Gaia didn’t have its eyelid (I may be wrong).
Ganmons of Giant Robo, however, had its eyelid and it was forced to be closed by Robo with his reacher-like equipment coming out of, if my memory is correct, his buckle before being beaten by him.
Actually we had another eyeball monster which appeared in Masked Ninja Akakage and is called Hitotsume (single eyed).
This is a monster regularly summoned by a villain ninja through his yojutsu (mysterious skill), and, as Akakage and Giant Robo were produced by the Toei Company, Ganmons might have been a developed form of Hitotsume in terms of the idea.
It is fully thinkable that the Tsuburaya people could have created Gan Q inspired by Ganmons although I don’t think they publicly referred to it.