In Homer, no. In other contexts, yes.
The classic passage for the Homeric Aithiopes is Odyssey 1.22-24:
But [Poseidon] had travelled to the Aithiopes, who are far away:
the Aithiopes, who are divided in two, the farthest of men,
some at Hyperion’s setting, some at his rising …
And in Odyssey 5, when Poseidon is on his way back from his holiday, he comes via the Solymoi mountains in Lycia, that is, travelling westward (Od. 5.283).

The earliest use of ‘Aithiopian’ that refers to the region south of Egypt and the Maghreb is in Hekataios. FGrHist 1 F 326 (Stephanos of Byzantion s.v. Υσαεῖς, BNJ translation):
Hysaeis: A small and a large island of the Ethiopians (Αἰθιόπων). In his Periegesis of Egypt, Hekataios says that the island dwellers are Hysaitai (equivalent to Oasitai).
and F 327 (Stephanos s.v. Σκιάποδες):
Skiapodes. An Ethiopian (Αἰθιοπικόν) people, as Hekataios says in his Periegesis of Egypt.
Even here we can’t be 100% confident that ‘Ethiopian’ was the name that Hekataios himself used — that’s the problem with second-hand fragments — but it’s a fair bet. And there’s that famous vase depicting an Aithiopian warrior at Troy flanked by two Amazons, where the Aithiopian is depicted with Ethiopian facial features.

There’s no real hint of that in Homer though. As to the division into east and west, a good article by Dimitri Nakassis explains that as a development from a one-pole in-out model, seen in Hesiod’s Tartaros, where Night and Day meet each other coming in and going out of the gates of night and day in a single place, towards a two-pole east-west model where the two poles are still closely linked. He uses the islands of Kalypso (utmost west) and Kirke (utmost east) as a paradigm, pointing out how Odysseus and his men are unable to find the sunrise or sunset while they’re there (Od. 10.190-192).
https://hcommons.org/deposits/item/hc:16737/

This mirrors another aspect of mythological thought, that getting to the Ocean (the river that surrounds all lands) gives access to all regions bordering on the Ocean, no matter which direction they’re in. When the Argonauts are trying to get home from Kolchis, some reports have it that they sail up the Tanais (the Don), northeast out of the Black Sea, get to the Ocean, and so are able to sail into the Mediterranean via the gates of Herakles (Gibraltar), in the west. It seems to be wrong to understand this as meaning they followed the Ocean westward until they got to Gibraltar: it seems more that once they were there, they were in a kind of mythological junction point.

It may be that that’s how the name Aithiops came to be transferred to Africa south of the Maghreb, since that too is imagined as bordering on the Ocean. https://aeon.co/essays/when-homer-envisioned-achilles-did-he-see-a-black-man