I have no doubt that Noxus under the trifilax rule would be able to take Ionia easily due to Swain's knowledge, Darius's legion, Le blanc's subterfuge. Also taking into account the fact that they have Sion as a living battering ram. The only issue that would impede their conquest would be Zed's order of shadow which includes Kayn and Rhaast and maybe Varus if Val and Kai forcehim to defend Ionia.
Demacia would be more difficult for Noxus to take due to the sheer amount of magic they have within their ranks which would basically permanently activate Galio until he is either destroyed or Noxus is driven back. Also their is the dauntless vanguard which is supposedly a small but very elite fighting force. I think Noxus would be able to take Demacia if their phobia of magic gets in the way of them fighting, such as them viewing galio as magic and attacking him or them persecuting lux and their other mages even though they could help. If not for that then Demacia wins due to simply having home court advantage and their petricite laden society impeding the likes of Swain and Leblanc in using their magic to disrupt Demacia. Of course any engagement outside of the walls of Demacia is a guarenteed win for Noxus in my eyes due to only piece of petrecite being present being galio which can be taken down with combined efforts of noxians and sion.
Shurima is basically guaranteed due to it being a fallen civilization that is way past its prime. Its just a bunch of fractured tribes that have no orginization. Only issues would be the Sai Khaleek since its a big area of death thanks to Rek Sai and maybe the ascended beings like Azir, Nasus and Xerath (Renekton would only be a problem if they run into him or he regains his sanity).
The Freljord cant be completely conquered due to the sheer amount of power that lies within that land. The Noxians would have the contend with the Winters Claw and the Ursine. Volibear and Anivia would almost certainly fight since in Volibear's case he is a warmonger and Anivia since she wants to defend freljord. Ornn might become a factor if he decides to fight or if the Noxians annoy him enough. I doubt Noxus has nearly enough resources to take down the demi gods. Also there is the matter of the Frozen Watchers and I severly doubt that if Iceborn couldnt kill them than Noxians certainly cant. Only chance of them winning is if the war within the freljord weakens every faction a considerable amount and Noxus approaches while they are vulnerable.
Noxus might have a tough time taking over Piltover at first due to its isolated location and its hextech but since Noxus has vast armies and their soldiers would never back down in attempts to emulate Kled's courage they would eventually overpower them through sheer numbers alone and their elite warbands. And I dont think that besides Camille can really disrupt the Noxian war effort in any meaningful way. Now Zaun on the other hand might be more of a challenge and might not even be able to conquer it due to all the wild cards within with Urgot and his followers causing chaos all around and since Urgot is familiar with Noxian tactics so he could probably disrupt their war efforts.
Warwick would probably fight for Zaun's people which means now Noxians have to deal with a chimeria man who has regenerative properties. Then you have the Viktor and his cult of technogically enhanced people which means more hextech will be used against them. This along with Janna which is a sort of pseudo godess means that Noxus has the weather against them and are fighting in unfavorable conditions.
Noxus would not have a chance in taking Mount Targon due to its harsh conditions which the people who inhabit it are accustomed too while the Noxians would not be. Also the likes of Leona and maybe Diana would come to defend their people which means that Noxians would have to deal with the aspects as well which doesnt seem like a battle they are winning.
The shadow isles is a another region that I just dont see Noxus conquering due to the mist killing anything it touches and turning it into mindless ghouls with only the intention to kill. Also there is the fact that there are wraiths such as Hecarim, Thresh and Karthus which would definitely see them as just souls to claim.
All in all I think Noxus could take a good chunk of Valoran but would certainly have issues invading some of the more distinct terrains such as Zaun due to its odd geography or Demacia due to its petricite walls.