In Startrek TOS Episode "Bread and Circuses" we see a planet with Roman Governance and gladitorial games.
In the Startrek Canon - we meet the Romulans, a race of people with a Roman governance model.
What is the relationship between these two?
(Only hypothesis I have heard is that the Roman gods were a several thousand year old advanced Alien race that visited all three planets and influenced them to varying degrees).
There isn't one.
Gene Roddenberry had an interest in ancient Rome, so it was simple coincidence (bolding mine):
Paul Schneider modeled the Romulans on the ancient Romans, naming the species' homeworlds after the mythical founders of Rome, Romulus and Remus. "It was a matter of developing a good Romanesque set of admirable antagonists that were worthy of Kirk," Schneider related. "I came up with the concept of the Romulans which was an extension of the Roman civilization to the point of space travel, and it turned out quite well." (Captains' Logs: The Unauthorized Complete Trek Voyages, p. 34) D.C. Fontana reckoned that Schneider basing the aliens on the pre-existing Roman civilization was the cause for the writer receiving insufficient credit for creating the Romulans. ("Balance of Terror" Starfleet Access, TOS Season 1 Blu-ray) Gene Roddenberry, interested in ancient Rome himself, approved of the initial depiction of the Romulan species. "He loved Paul's having endowed the enemy-Romulans with the militaristic character of the ancient Romans," wrote John D.F. Black and Mary Black. (Star Trek: The Magazine Volume 2, Issue 11, p. 19)
The "extension of the Roman civilization" part can only be interpreted as "this is how their society is modelled", not a literal extension of our Romans, because Romulans were never human.
The theories you've heard probably stem from mixing up the Greek and Roman gods, as one Greek god, Apollo, does appear in TOS 2x02, Who Mourns for Adonais?. The Romans seem to have adopted Apollo as their own god as well, with the same name, but within the episode he identifies himself as Greek.
However, in-universe we have Hodgkin's Law of Parallel Planetary Development which means we don't need Roman godlike aliens to explain away the similarities.
I'm not sure there is any official/cannon relationship but there is plenty of potential for a good connection.
Hodgkin's Law explains the the planet in "Bread and Circuses". It's pretty much an exact parallel of Earth where Rome never fell.
It would make perfect sense if the vulcans that left to colonize space encountered that "Bread and Circuses" planet during their travels and just fell in love with the Roman culture so much that they incorporated it into their own culture and customs, started calling themselves Romulans, etc.
When they finally found a system with an inhabited planet perfect for the conquering and a second planet just waiting to be colonized, they named the planets Romulus and Remus, labelled the inhabitants of Remus "Remans", made them their slaves and created a new Empire with many "Roman" parallels, etc.
That's just my theory but I like it.
It is possible that Romulan society is a mixture of Vulcan-descended colonists and ancient Romans brought to the planet by the Preservers.
Of course if any Romulans are Humans descended from ancient Romans they haven't been seen or mentioned onscreen.
Remember that "Bread and Circuses" (written by Roddenberry and Gene Coon) was about a culture that remained culturally "frozen" (more-or-less) while advancing technologically to the equivalent of 20th-century Earth, only to find their way to a monotheistic religion including a "Son-of-God" figure around the time Enterprise visits there. Or, you could say it as: "what if Jesus only appeared in our mid-20th century?" The fact that it resembled ancient Rome has more to do with the cultural context in which Jesus lived (Roman empire, not Rome itself).
It would seem that both "Balance of Terror" (where the Romulans are first introduced) and "Bread and Circuses" are both reflections of Roddenberry's fascination with ancient Rome and nothing more.
