The foundation of the “white gods” narrative in popular history today (for example, in Guns, Germs, and Steel) stems from the story of Cortes being supposedly mistaken to be Quetzalcoatl by Montezuma. When investigating the origin of this legend one will find that the only contemporary source of the conquest of Mexico, Cortes’s letters to Charles V, neglects to mention the natives worshiping Cortes as a god of any kind. In fact, the letters from Cortes provide clear evidence that Montezuma believed the contrary:

‘I know they [the Totonacs] have told you [Cortes] also that I have houses, with walls of gold and that the furniture of my halls, and other things of my service, were also of gold, and that I am, or make myself, a god, and many other things. The houses you have seen are of lime and stone and earth.’ And then he held up his robes, and showing me his body he said to me, ‘Look at me, and see that I am flesh and bones, the same as you, and everybody, and that I am mortal, and tangible.’^1 [emphasis added]

According to both Camilla Townsend, author of Burying the White Gods, and Matthew Restall, author of Seven Myths of the Spanish Conquest, the Quetzalcoatl as Cortes myth does not exist in any contemporary primary sources, rather it originates in Bernardino de Sahagun’s Florentine Codex in the 1560s several decades after the Spanish conquest of Mexico.^2 ^3

Besides the motives of Sahagun to present the Spanish conquest as divinely ordained in order to convert the native population, the story itself is very inconsistent. Quetzalcoatl was not a major god in Tenochtitlan and not the patron god of Montezuma or the Mexica; that was the war god Huitzilopochtli. Quetzalcoatl was the patron deity of Cholula which was one of the only Mesoamerican city-states to attack Cortes; this is an inconsistency as Cholula would not attempt to kill its patron deity.^4 Furthermore the notion that Cortes appeared on the year of the prophesied return of Quetzalcoatl is false. The same Florentine Codex which supports the idea also says that the conquistador Grijalva was supposedly mistaken to be Quetzalcoatl, yet he arrived in Mesoamerica in 1518 while Cortes landed in 1519.^5

But I realize you were asking about Bernal Diaz. What does this all have to do with the Quetzalcoatl myth? Bernal Diaz’s History of the Conquest of New Spain was written decades after the Florentine Codex; by then the myth of Spanish deification was already widely spread. Perhaps Sahagun’s influence could explain why he so often makes references to the natives calling the conquistadores “teules.” However I can only correlate the timeline between the two, there is not hard evidence (that I know of) to prove a connection. At first Bernal Diaz understands the word to mean either god, saint, or demon.^6 However Diaz only makes this distinction once in his memoirs and afterwards he always defines it as meaning “gods”.^7

But the word teules does not exist in any Mesoamerican language; it is a Castilian corruption of the Nahua word teteoh (plural), and teotl (singular).^8 The concept of teotl in Aztec theology is much different than that of saints, demons, or God in Christianity. Things with the quality of teo (root form) are very diverse, ranging from the mundane such as maize and water to deities who imbued things, such as maize and water, with their divinity. ^9 Things attributed with teo were associated with divinity but were certainly not always deities themselves. For example maize itself was not a god just because it was a teotl, but there are multiple gods of maize who are all teteoh. Other things possessing the qualities of teo were mighty predators such as jaguars and people such as human impersonators of deities and powerful tlatoani (kings); so it is not extraordinary that the conquistadores would be called teteoh.^10 However the labelling of the Spaniards as teteoh was not equivalent to calling them gods. The term is so multifaceted that it is possible for the natives to be calling the Spaniards “deity impersonators” in the sense that they are potential sacrifices for the Aztecs.^11 People who impersonate the gods by dressing in their effigies are teteoh, which may perhaps be why Diaz notes the perceived divinity of a conquistador's helmet by Montezuma. Regardless all Aztec would know that a god impersonator is not actually a god; they just possess aspects of divine power, or teo (as many things in the world do). They would notice likewise with the Spanish.

Furthermore in her essay Camilla Townsend argues that the widespread use of the word teules to refer to the Spaniards was primarily because they had no other term to refer to the Spaniards by. “The indigenous had to call the Spaniards something, and it was not at all clear what that something should be.”^12 Everyone in Mesoamerica already came from a place that was known and place that there were words for but the Spaniards did not. Much like how the Spaniards mistakenly referred to the Native Americans as indianos the Mesoamericans referred to the Spaniards as teules simply because that is what they called themselves after it started happening. After the conquest the natives were given different ways to refer to the Spaniards, such as cristianos, but in 1519 the Spaniards were novel to (most) Mesoamericans.^13 Which is why the labelling of the Spaniards as teules is not an indication that the natives (well, at least the Mexica) thought the Spaniards were gods.

In conjunction with calling themselves teules, the conquistadors intentionally attempted to play off the superstitions of natives in order to be viewed as teules. They worked toward this end with elaborate tricks such as burying their dead in secret to make the Cholulans think that they were invisible. However even these tricks were not typically effective; in the case of the Cholulans they kept attacking the conquistadores anyway regardless of whether they were teules. They were eventually defeated by Cortes and large army of Totonac allies. Whenever these tricks were effective, such as when Cortes intimidated the Tabascans into making peace by firing cannons and using horses, they can almost always be attributed to some kind of shock factor or technique that was new to the natives, such horses, cannons, and guns, and not indicative of a belief in Spanish godhood.

As this answer is already very long, I'll stop here. But if you are interested in the "prophecy" in Diaz's memoir where Montezuma gives a speech to Cortes that seems to support the Quetzalcoatl myth, I can post the rest of the argument I cannibalized for this post.

TL;DR: The myth of the white gods is not supported by a critical analysis of primary sources and doesn't make sense in Aztec theology anyways.

Sources:

1- Cortes, Hernan. Letters of Cortés: The Five Letters of Relation from Fernando Cortes to the Emperor Charles V, trans. MacNutt Francis Augustus. New York and London: G.P. Putnam's Sons, 1908. Page 235.

2- Townsend, Camilla. Burying the White Gods: New Perspectives on the Conquest of Mexico. The American Historical Review 108 (3). Oxford University Press, 2003 Pages 659-87.

3- Restall, Matthew. Seven Myths of the Spanish Conquest. New York: Oxford University Press, 2003. Page 113.

4, 5- Townsend, Camilla. 668.

6- Restall, Matthew. 112.

7- Townsend, Camilla. 672.

8- Restall, Matthew. 112.

9- Townsend, Richard F. The Aztecs. Third ed. London: Thames and Hudson, 1992. Page 120.

10- Townsend, Richard F. 120.

11- Townsend, Camilla. 672.

12- Townsend, Camilla. 670.

13- Townsend, Camilla. 672.

Also, lots of credit is due to /u/400-Rabbits whose arguments made in various posts on this sub I made used as a base for this writing. I believe he has several answers on this topic of Spanish godhood, if you're interested.

There's always room for discussion, but perhaps the section Spanish "Gods" from our FAQ will answer your inquiry.