Are the words “fact” and “fairy” etymologically related?

Bruno Latour remarks in a 2003 lecture that “fact and fairy are etymologically related but I won’t develop this point here”.

We can summarize, I estimate, 90 percent of the contemporary critical scene by the following series of diagrams that fixate the object at only two positions, what I have called the fact position and the fairy position—fact and fairy are etymologically related but I won’t develop this point here. The fairy position is very well known and is used over and over again by many social scientists who associate criticism with antifetishism. The role of the critic is then to show that what the naïve believers are doing with objects is simply a projection of their wishes onto a material entity that does nothing at all by itself… Is it not time for some progress? To the fact position, to the fairy position, why not add a third position, a fair position?

_{Why Has Critique Run out of Steam? From Matters of Fact to Matters of Concern}

According to Wikipedia, the word fairy has several variants: fay, fae, fair folk; faery, faerie and is derived from the Latin fata. The suffix -erie (-(e)ry) was added to faie to become faierie, and signified the land where Faes dwelled. Whereas fact is derived from the Latin factum.

But what is the connection between fact and fairy that Latour mentions?
Are there any leads on this?

No, they're not related. Both do come from Latin words, but not from the same Latin words.

Fairy, or faerie, is a French word derived from fé, a descendant of the Latin word fidēs, 'faith'.
Here's what the OED says about it:

The L. fidēs, like its etymological cognate Gr. πίστις (pístis),
which it renders in the N.T., had the following principal senses:
1. Belief, trust.
2. That which produces belief, evidence, token, pledge, engagement.
3. Trust in its objective aspect, troth; observance of trust, fidelity.

Fidēs in turn comes from the PIE root *bʰeydʰ 'to trust'.

Fact, however, is from factum, a form of the Latin verb facere 'make, do, accomplish'.
Originally, fact in English meant 'a thing done or performed; action, deed, course of conduct'.
Nowadays we're more likely to use fact about states and abstractions, but originally it referred to events and actions only.

Facere comes from the PIE root *dʰeh₁ 'to do, put, place'

Interestingly, the etymological roots do still distinguish the two senses. Fairies are imaginary,
and depend on human belief, while facts are actual, and have no need for belief; they just are.

Fact and fairy aren't etymologically related, but John Lawler's answer is probably not correct when it states that fairy is related to Latin fidēs. According to regular vowel changes, fidēs corresponds to the French word foi 'faith'. The Old French form fei, which is the source of an obsolete English word "fay" meaning "faith", looks a bit more similar to fairy in form, but as far as I know was not used with a "fairy"-related meaning.

The usual etymology given for French fée 'fairy' (in Old French also spelled fae or faie) is instead that it is related to Latin fātum 'fate, doom' via the plural form fāta, which you mention. This plural form is supposed to have been reinterpreted as a singular feminine noun, which was a common development for Latin neuter plural forms. Regular Latin-to-French sound changes get us from fāta to faie/fae/fée.

Fātum/fāta in turn is supposed to come from the Latin verb for 'speak', which Wiktionary says is from a PIE root that is reconstructed as *bʰeh₂-. This only shares the first consonant in common with *bʰeydʰ. (The vowel "e" included in the citation forms of most Proto-Indo-European roots is typically not a distinctive part of the root, but can be replaced according to the system of "ablaut" or "vowel gradation"). And it has nothing in common with the root *dʰeh₁.

The path from fātum to fairy is somewhat complicated; I go over the hypothesized intermediate steps in my answer here: Etymology of “fairy”