Negative. (Or I should say: not always.)
Cardassia Prime, source-world of the Cardassian race (as well as the seat of government of the Cardassian Union), was the second planet of it’s system.
Edit: other places list it as the sixth planet. Sigh.
Rigel IV indeed occupies the fourth orbit around its star, but I find no mention of a Rigel Prime. In cumulative Trek lore, the Rigel system is actually home to four different intelligent species on four different planets–III, IV, V, and VII–but none of them appear to have expanded to other worlds. Presumably this combination precludes any of the system’s planets from acquiring a “Prime” designation.
The orbital position of Tellar Prime is not clear, but it is certainly the homeworld of the Tellarites. Sources may dispute the orbit of Cardassia Prime, but none of them put it closest to its star.
In Trek, “Prime” is pretty clearly a designation of rank/importance, and Roman numerals are used (by the Federation) for orbital position.
The cast of “Number of the Beast” used a similar convention to denote their own home space/time. They did state this was arbitrary on their part. 
In Chronicles of Riddick, much of the action takes place on Helion Prime. ‘Prime’ here means both most important and closest to the system’s sun.
Dune also has ‘Salusa Secundus’, so Frank Herbert at least did appear to just be thinking of planetary order.
That’s funny. Salusa Secundus is the 3rd planet in its star system (Gamma Waiping). I’m not sure Secundus is supposed to be a reference to 2nd.
In Blake’s 7, Gauda Prime was the location for the final episode.
Switching formats a bit, *Eden Prime *is a location in the Mass Effect game universe. From the game wikia, it looks to be the second planet out, so that makes the “Prime” designator another vote towards marking either settlement time or importance to the namers (in that game, Eden Prime is one of the first exoplanet human colonies) rather than a mark of orbital location.
I should have known better, since this is Star Trek after all, but I just picked Rigel randomly.
Re: Cardassia Prime. I always assumed it was supposed to be the closest to its star when it was named (it is presented as being hot) and references otherwise were mistakes but so be it.
shrug Secundus means second in latin… Maybe it was supposed to be the second planet in the system that was colonised? It doesn’t have to be a straight ‘number of planets from the star’.
Alternatively, it does mention that it was the ‘second stopping point in migrations of the Wandering Zensunni,’ though that would be an odd thing to name something after in anything but a story.
Since this story came out in 1944, it predates all the other examples by quite a wide margin (though it may not have been the first use of the convention).
But Brown didn’t use the word “Prime.” The known planets there are called Thor and Freda, or Sirius I and Sirius II.
Here’s a list of planets in Star Trek designated as ‘Prime’.
The handful that note their orbital position are not the innermost planet. Most of them are inhabited (either species homeworlds, or colonies). It’s pretty clear that ‘Prime’ does not mean the innermost planet (there wouldn’t be such a huge number of inhabitable - let alone inhabited - Primes, if it did).
The innermost planet in a system is designated as I, not Prime.
Prime indicates, as far as I know, the homeworld for a species or family.
Phoenix Prime was a fave of mine back in the day. Especially the Frazetta cover…
Also, Heinlein mentions Vega Prime in Citizen of the Galaxy (tho referring to a starship) and Tellus Prime in to Sail Beyond the Sunset.
Nice finds Qadgop. Heinlein predates Herbert’s use, so maybe Heinlein started it!
I think, Dearest OP, that if you seek the first example of the term ‘Prime’ in Science fiction ~ you’ll have to define the genre- for your needs, in any case.
Some say Gilgamesh or Beowolf were Science Fiction, or had Science Fiction Elements. The Arabian Nights
Included Tales of Robots.
Some say Frankenstein. {1818}
Some say Jules Verne & Wells.
So, where does your Quest begin? with what Novel, or from what date?
It drives me a little nuts that they do this. Why would the Cardassians call their world Cardassia Prime? We’re humans from Earth, the Klingons are from Qonos (although at one point the writers evidently intended them to be from Kling), but the Cardassians are from Cardassia Prime. Wouldn’t they have some historic name for their homeworld that predates their having colonized other planets? Wouldn’t the name for their species likewise not be a variant of their planet’s name (since it would have served the purpose of differentiating them from other Cardassian species)?
My fanwank is that they are actually saying a different name in their native language, but the universal translators render it as Cardassia Prime (like translating Deutschland as Germany). My double-fanwank is that it’s rendered as something else altogether that then gets dubbed into the English equivalent for the benefit of the viewing audience.
Of course, then Terran should really be translated as human, since we don’t call ourselves “people from terra”. To say nothing of the inexplicable ferengi word hoomon.
I think their historical planetary name is “Cardassia,” and often that’s all they say, as in the Bajoran-sailship episode (which references a time before they settled other worlds). The “Prime” comes in when they speak of it specifically in the sense of its original/central status, the seat of government, so on.
Babylon 5 also used Prime, evidently for the homeworld of each race. Centauri Prime is mentioned in almost every episode, and Drazi Prime is mentioned two or three times in the fifth season. Can’t think of any others right off hand.