To the best of my knowledge, your question is not answered in any canon source. Obviously I may be missing something, and I haven't read season 10 of the comics, but here's what I know.
The only depiction I could find of the actual first slayer (as opposed to a spirit or illusion) fighting anything is in the comic book Tales of the Slayers. She appears in the beginning fighting (and slaying) a vampire. This could possibly be a Turok-Han - it's not possible to tell from the illustration. She refers to it only as "the beast", while the girl she speaks with afterwards refers to her as fighting "the vampires".
However, as this was published in 2001 (season 5 was 2000 - 2001) while the Turok-Han was only first mentioned in season 7, I doubt the Turok-Han had even been conceived of when this comic was written. (Not that retcons are out of the question.)
Beyond this, I can only speculate based on what little evidence we have.
Here is Giles' full quote:
Uh, what you fought was a vampire, but it was, um, something more than
that. It was a-- a Turok-Han. As Neanderthals are to human beings, the
Turok-Han are to vampires. They're primordial, ferociously powerful
killing machine, as single-minded as animals. They are the vampires
that vampires fear. An ancient and entirely different race. And until
this morning, I thought they were a myth.
No Turok-Han have been seen in long enough time that Giles believes they are a myth. So presumably there is no first-hand written account in the Watchers Council records. However, the first slayer predates the Watchers Council, and possibly written language - or at least, maybe the shadow men didn't keep written records or they were lost. "Primordial" does suggest prehistoric to me, though that may be poetic license.
The episode the first slayer first appears in is called "Primeval", which suggests she is similarly ancient, but these are vague descriptions.
Comic Vine's description of "the primitive" (as the first slayer may be referred to) states she lived around 5000 years ago, which would put her in the bronze age. However, I don't know what their source for that is and wouldn't rely on it.
I'm going to venture into hypothetical territory and guess the Turok-Han disappeared from the earth around the same time many of the other demons did. Given their feral and vicious nature, they probably couldn't hide among humans the way modern vampires do - I would speculate this is why they didn't survive on earth to the current day. Perhaps the slayers themselves contributed greatly to the extinction of the Turok-Han.
Assuming my hypothesis is correct, and given the demons had (presumably) not been driven from the earth (or into hiding) when the slayer was created, it is likely there were still Turok-Han roaming the earth during the time of the first slayer. In which case it seems likely to me she would have fought one at some point.
Even if she did fight Turok-Han, it is possible modern vampires existed at the time as well. No way to know for sure.
One counterpoint is in the episode "Get It Done" (S7E15), when Buffy meets the shadow men. They show her a vision of the Turok-Han beneath the hellmouth. It is possible they were already there at the time of the shadow men. However, the shadow men may simply have had some ability to see the future (as the slayer does). I think the latter is more likely. I'm not even sure Buffy was actually transported to the past - it could have been some other dimension.
