I don't think that Romans ever referred to "Celts" (which was a Greek word). "Gauls" however encompassed (probably) Celtic-speaking populations (but some groups may have differed linguistically) in Gaul, Northern Italy and the Danube Basin. AFAIK, it did not extend into the British Isles or all the Celtic-speaking parts of Iberia. So there were groups who spoke languages we now consider Celtic, which weren't associated with Roman Gauls by the them. I think the use of "Celts" in Greek sources mirrored closey the use of "Gauls" by Romans.
However, they both though in terms of kinship, not language.
Celtae and Celtica were both used by Romans to describe the Gauls and Gaul. The word "Gaul", which is of Germanic etymology, the same as Wales, Wallachia, and Walloon, was never used by Romans. Gallia was, however, though it's etymology is unrelated to the word Gaul, and likely to be related, in fact, to the word Celtae. Also, Gauls referred to themselves as Celtos or Celti. So there's that.
Celtic identity is tricky however. If we were to go by Roman designations (which is dangerous at best), you'd have British, Gallic, Belgic, Iberian, and Helvetian, Boiian, Hibernian, Noric, and Galatian all related to each other within what we would designate as Celtic. They all seem to have shared a related material and linguistic heritage, though their genetic relationship varied. Note most Irish DNA is more closely related to those of the Basques than those of, say, the Bretons.
To the OP: the simple answer is YES, the Goths are related to the Gotlanders, and shared a common ancestor.