As you probably know, the Ethiopian claim centers on the notion that the Queen of Sheba bore Solomon's son, Menelik, after returning to her homeland. The tradition dates to at least the fourteenth century C.E. and perhaps as far back as the sixth. But none of these details appear in the Biblical narrative (1 Kings 10:1-13), and nothing there suggests that Solomon and the Queen engaged in sexual relations.
The historical Sheba/Saba actually lay in south Arabia (today Yemen), and the assumption that the Queen of Sheba came from Ethiopia possibly stems from a conflation of the "queen of the south (i.e., Sheba)" in Matthew 12:42 and Luke 11:31 with "Kandake Queen of the Ethiopians" in Acts 8:27. However, as Bar Kribus recently pointed, the Ethiopian Kingdom of Aksum did, at times, exert influence or control over the Sabaeans (and eventually their conquerors, the Himyarites). Insofar that the Aksumite kings presented themselves as the rulers of Saba, the Ethiopians may have a genuine if rather distant connection to the land of Sheba.
Still, the historicity of the biblical tale of the Queen of Sheba remains impossible to determine in the absence of external evidence. But we do have evidence of ancient Arabian women exercising some degree of power and independence, and trade between Arabia and the Levant appears to have already existed in the time of Solomon (tenth century B.C.E). In fact, the seemingly out-of-place reference to the Tyrians (Israel's trading partners) and Ophir (a source of gold, almug wood, and precious stones in Arabia) in 1 Kings 10:11 alludes to a commercial context for the meeting between Solomon and the Queen.
Hopefully someone else can provide more information on the history of the Ethiopian tradition! :)
Sources and further reading:
General: Witold Witakowski and Ewa Balicka-Witakowska, "Solomon in Ethiopian Tradition," in The Figure of Solomon in Jewish, Christian and Islamic Tradition: King, Sage and Architect, ed. Joseph Verheyden (Leiden and Boston: Brill, 2013), 219-240; Bar Kribus, "Arabia or Africa: Where Is the Land of Sheba?," Biblical Archaeology Review 42, no. 5 (2016): 26-36.
Saba/Sheba: Israel Eph'al, The Ancient Arabs: Nomads on the Borders of the Fertile Crescent, 9th-5th Centuries B.C. (Jerusalem: Magnes Press, 1982); Kenneth A. Kitchen, "Sheba and Arabia," in The Age of Solomon: Scholarship at the Turn of the Millennium, SHCANE 11, ed. Lowell K. Handy (Leiden, New York, Köln: Brill, 1997), 126-153; André Lemaire, "La Reine de Saba à Jérusalem: la tradition ancienne reconsidérée," in Kein Land für sich allein. Studien zum Kulturkontakt in Kanaan, Israel/Palästina und Ebernâri für Manfred Weippert, OBO 186, ed. Ulrich Hübner and Ernst Axel Knauf (Freiburg: Universitätsverlag; Göttingen: Vanderhoeck & Ruprecht, 2002), 43-55.
Trade: Mario Liverani, "Early Caravan Trade between South-Arabia and Mesopotamia," Yemen 1 (1992): 111-4; John S. Holladay, Jr., "Hezekiah's Tribute, Long-Distance Trade, and the Wealth of Nations ca. 1000-600 BC: A New Perspective," in Confronting the Past: Archaeological and Historical Essays on Ancient Israel in Honor of William G. Dever, ed. Seymour Gitin et al. (Winona Lake: Eisenbrauns, 2006), 310-331, esp. 314-321; Michaël Jasmin, "The Emergence and First Development of Arabian Trade Across the Wadi Arabah," in Crossing the Rift: Resources, Settlements Patterns and Interaction in the Wadi Arabah, LSS 3, ed. Piotr Bienkowski and Katharina Galor (Oxford: Council for British Research in the Levant and Oxbow Books, 2005), 143-150.