Do we know, for sure, if Godzilla is a male?

Godzilla is always referred to as male and referenced with "he" or "him."

Do we know, for sure, if Godzilla is a male? Has beast gender ever been a factor within any kaiju movies?

Answer

Short answer: No, we don't. Mostly because the Godzilla series has no decent continuity.

Long answer:

In the original Japanese versions of the movies Godzilla and other kaiju are mostly referred to by the gender-neutral it. In the English translated versions they refer to Godzilla mostly as he, hence the confusion.

So basically they (the creators/trademark owners) don't really want to put a gender on Godzilla.
He has a son, however, who hatched from an egg. This again enhances the confusion about its gender, especially since it's never cleared up where the egg came from and if Godzillas son is actually his offspring or just another member of his species.

According to my Google-Fu the 1998 Hollywood remake movie is so far the only one to put a gender on Godzilla (asexual reproduction, egg-laying and all that). However, since the creature from that movie is officially no longer recognized as Godzilla, it' doesn't count.

Toho (the company which created Godzilla and made all the movies) apparently didn't like the 1998 depiction of Godzilla very much. Saying among other things, that they took the "God" out of Godzilla. This is why they created a new monster named Zilla for the 2004 movie Godzilla: Final Wars (marking the 50th anniversary of Godzilla) as kind of a mockery of the 1998 American version.
He's one of the monsters Godzilla fights in the movie and is defeated in what is apparently one of the shortest fights in Godzilla history.

In fact, the Hollywood Godzilla from 1998 has officially been renamed Zilla by Toho (check here for sources) and in all newer depictions of the creature (which is apparently still used) it's called Zilla. Only the 1998 movie itself still calls the creature Godzilla.

All the original Godzilla movies give no indication as to its gender. In fact, there are even several origin stories, some of which would allow for the eggs (from which the baby Godzillas hatch) to be artifacts of pre-historic times, just like Godzilla itself.

It should be noted that the Shōwa series movies (from the original movie up to 1975, including Son of Godzilla) featured no real continuity at all but are rather movies based on the successful first movie.

In 1984 Toho began the so called Heisei series with The Return of Godzilla and rebooted the Godzilla universe. Starting with this movie all but the very first Godzilla movie (which is the only one tying all timelines together) never happened. Toho established a continuity between the movies which was later messed up again by introducing time travel.

Baby Godzilla was also reintroduced later in the series, but once again only as an egg that is one day just found. Nothing on what Godzilla's actual relation to the baby is and also no information on the gender.

A common factor in all the different continuities (there is also the Millennium series after 1999 up to the 2004 movie) is, that there is more than one Godzilla. The Godzilla from the original 1954 movie died at the end of the movie (killed by a device which apparently couldn't be reproduced since the plans were destroyed as well) and originally no sequels were planned.
In the second movie it was specifically stated that Godzilla is dead and that this time Japan is under attack from a second Godzilla. In one of the later movies the first Godzilla's bones are even used as the core of Mechagodzilla which then goes on to fight the current Godzilla.

There are also several plot holes in the movies themselves. For example, if there have been at least two Godzillas, why did they end up with the exact same mutation? Or why did Godzilla mutate the exact same way during the time travel plot, when he was not irradiated by a nuclear bomb but by a soviet submarine some time later.

Combined with the general lack of continuity this allows for additional speculation. If there were at least two Godzillas at one point, one of them might have been male and the other female. That Godzilla has a son seems to indicate that there are indeed two genders. So the counter question to the question of Godzilla's gender might be: Which Godzilla?

In summary: A clear answer is never given and that is most likely deliberate. Godzilla is not really supposed to have a fixed background. It changes with the movies and for the movies, reflecting the message the film makers wished to convey. Just like he became a good guy and then turned back to be more of a villain. They wanted him to have a son to make him more family friendly (this was also during a period where Godzilla became friendlier in general, turning more and more into a kind of guardian). So they gave him a son from an egg and to this day nobody bothered to explain exactly the why and how.