If so, did the Aztec think he was that god? Did this prophecy play any role in the Aztec's fall or in the way they treated Cortés?
Ross Hassig doesn't think so. Cortes had been preceded by two other Spanish expeditions to Mexico, and the second, led by Juan de Grijalva, had been met by the Aztecs. From 'War, Politics and the Conquest of Mexico' in War in the Early Modern World 1450-1815, ed. Jeremy Black (1999):
Although the Aztecs had not determined who these white-skinned strangers were, this meeting undermines the widely-accepted interpretation that the Aztecs thought Cortes was the returning god, Quetzacoatl. Cortes landed in 1519, the Aztec Year 1 Reed, which was also the birth year of Quetzacoatl and thus, presumably, the year in which he might return. But Grijalva's arrival in the Year 13 Rabbit makes this interpretation less plausible.
If the Aztecs hailed Cortes as a returning god, why him, and not the other Spaniards who had arrived in Yucatan years earlier?
Nor was Cortes immediately welcomed. When Cortes arrived, he was initially able to ally with the Totonacs, but then proceeded to Tlaxcallan, where he was opposed. The Spanish were able to hold off the native troops, but their small number meant they were on the verge of being overwhelmed before they were able to negotiate an alliance with the Tlacaltecs. Spanish accounts naturally present this as a result of Cortes' superior skill at diplomacy, but Hassig argues that it was a Tlaxcalan initiative when they noted the benefit Spanish soldiers could grant to their own army--they had recently been forced to acquiesce to Aztec hegemony and saw the Spanish as helpful tool with which to retaliate. Hassig characterises the entire 'Conquest of Mexico' not as a Spanish-led conquest but as an internal conflict between warring Aztec city-states in which the Spanish were co-opted. Despite their small numbers, he argues, their steel weaponry, tight formations, guns and cavalry had a decisive shock effect on native troops who tended to fight in dispersed formations. Mesoamerican battles tended to involve forces broadly equal in numbers, weaponry and tactics, so they usually descended into an extended slog with neither side's battle line breaking easily. However, the Spanish could create a breach in the enemy line with relative ease and this could be exploited by the far larger numbers of native troops and result in decisive victory. The Spanish, who barely numbered over 1000 and often far less, were accompanied in their battles against the Aztecs by over 100,000 native allies. They were able to make it past Tlaxcallan not because Cortes was welcomed as a god, but because they were a useful military force. 'Alone, the Spaniards could only hope to fend off attackers, but this alliance multiplied the effectiveness of the Tlaxcaltec army.
The 'returning god' explanation is often cited as evidence for why the Spanish were able to enter the Aztec capital at Tenochtitlan and apparently take up a position from which they could control the Aztec king Moctezuma, launch a pre-emptive strike in which 8-10,000 Aztec nobles were massacred, and eventually (probably) kill Moctezuma when he had served his purpose. In hindsight this was a mistake by Moctezuma, but there were good reasons he would welcome Cortes into the city quite apart from any idea that he was a god. Cortes arrived in the agricultural season, when the men of fighting age were busy with the harvest; it was not possible for Moctezuma to raise anything like enough men to fight a war, particularly since Cortes had by that point allied with the Tlaxcaltecs. Moreover Cortes was a new and unknown quantity which in the extremely decentralized Aztec empire could mean a rapid re-shifting of allegiances, as the Tlaxcaltecs had already done; Moctezuma had no desire to spark a civil war by strongly opposing the Spanish until he had properly assessed the political ramifications of their arrival and, more importantly, was able to raise an army. With Cortes inside the city he could be supervised and quickly dealt with if he turned out to be a threat. Thus there are far more mundane factors which indicate why events progressed as they did without us having to resort to assuming the Aztecs saw Cortes as a god.
By the way, there's an AMA tomorrow/today (the 14th) on 'Pre-Columbian civilizations of the Americas', in which our Mesoamerican experts can surely give you a more detailed answer than I can; but I hope this helps for now.
I'm not trying to discourage further discussion on this topic, but I have written a response to this question before. To give you a brief summary, we don't know, but it is probably a myth that arose shortly after the conquest. Or to be more precise, it is probably a series of factual events, misunderstandings, and outright myths that were stitched together in the early colonial period as part of a strategy to convert the natives to Christianity. Either way, the evidence that Cortés was seen as an incarnation of Quetzalcoatl is very shaky and highly suspect.