The one term we do know local Slavs from today's western Slovakia and Moravia used for themselves was Sloven, or in a spelling variation, Slovien. These two are all we know. Did they call themselves Great Moravians ? Unlikely. Moravians when referring to their realm ? Maaaybe. Did those that lived in the former Nitra principality in western Slovakia refer to themselves as Nitrans ? Not very likely, outside of maybe the town of Nitra and Nitra river region itself, but some might have understood they were being ruled from Nitra, and probably at least knew they are the subjects of this or that prince that was ruling at the moment. (Great Moravia only had one dynasty, really, the "Mojmír dynasty", as historians have nicknamed it, given that the "Pribina dynasty" was outcast from the Nitra half and went into exile down to Lake Balaton.)
Fun fact: Slovak and Slovenian, though now in the western and southern branches of Slavic languages, share some ancestry. Back in the Nitra and later Great Moravia days, some of Slavic tribes inhabiting today's western Hungary were broadly related to the Nitra principality's more consolidated tribes to the north, and the Slavic tribes to the southwest, in the area of today's northeastern Slovenia and southeastern Austria. The language obviously wasn't entirely the same, due to geography alone, but the dialects that stretched in this area formed a certain continuum of early Slavic dialects. As a result, even modern Slovenian is still distinctly unlike Serbo-Croatian and other south Slavic languages, and does have some holdovers and phonetics that make it sound closer to Slovak (or even some Moravian folk dialects), while still being a distinctly different language that's not easily mutually intelligible.
The only idea we have about the demonym of the Slavs from the Nitra principality and Great Moravia (after Nitra got absorbed into GM) is that they were called either Sloveni or Slovieni. Nothing else is known. But given the peculiar linguistic history connections between Slovaks and Slovenes - albeit these are connections more than a millennium old, long before either of the nations really formed - it's not all that surprising. It was probably a common word some of the western Slavs used to describe themselves as a people. (The term itself might be ethnocentric, related to slovo, "word", in the sense of "those whose words (language) we can understand". Part of why Teutons and later Germans were dubbed Nemci/Niemtsi, etc. by some Slavic nations - "mute ones" - referring to the fact that the Germanic languages weren't readily intelligible to Slavic language speakers, unless they bothered to learn them of course.)
How do you say a Slovak woman in Slovak ? Slovenka. How do you say a Slovenian woman in Slovenian ? Slovenka. What about the fellas ? Slovák and Slovenec. Hm ! How would they call something Slovak or Slovenian in their mother tongues ? Slovenský and slovensky. Oops ! Again !
What would a Slovak call a Slovak, the language, the country ? "Slovák, Slovenka, Slováci/Slovenky, slovenčina, Slovensko."
Okay. What would a Slovak call a Slovene, Slovenes, their language and the country of Slovenia ? "Slovinec, Slovinka, Slovinci, slovinčina, Slovinsko." Clear distinction there, even if still close.
What do the Slovenes of today call themselves ? "Slovenec, Slovenka, Slovenci, we speak slovenščina, the country's Slovenija."
Note the single-letter difference between slovenčina and slovenščina. And all the other close similarities. This isn't a complete coincidence. Some of the Slovak and Slovene ancestors from a thousand years ago used languages from the same continuum, at least as it was then. Before 1000 AD, the linguistic differentiation of Slavic-speaking peoples into western, southern and eastern branches was still ongoing, and the edges between the three major branches were often still fuzzy and had not undergone further drifting from each other and deeper differentiation.
I'd say it's a similar parallel to how you could take archaic proto-German spoken by the Saxons of continental Europe, compare it with the early medieval Saxon language spoken by the Saxons that migrated to the Low Countries and then the British Isles, and you'd still find some similarities. After all, even modern English still preserves terms like hound (comparable to German Hund, a dog in general), father (Vater, or Dutch Vader), etc., etc. There is semantic shift, of course. This has occured between Slovak and Slovenian as well, given how long ago they diverged from the same early Slavic "dialect pool", but you can still find similarities, some even pretty striking.
And what would the Slovenes say about Slovakia, after rubbing their chins thoughtfully ?
"Ah, the country of Slovaška ! Hello, Slovaki friends !"
"Um, really ? Sounds a bit different..."
"Yeah ! You speak slovaščina !"
"Hm... We are Slováci. And you speak slovinčina."
"The hell is slovinčina ?! We speak slovenščina."
"Are you sure, Slovinci buddies ? Slovenščina ? Sounds like our slovenčina."
"Argh... Let's just agree that both our demonyms come from a generic term for "Slav". "
"All right. And the ladies are still Slovenky ! Yay !"
"Yeah, but without the "y"."
"Oh ! Sorry."