What is the process called to change "fire" → "fiery"?

It's clearly not "conjugation", and I'm not even sure which keywords to use for google to help on this. Without having time to dedicate my next few days to read though linguistics textbooks, I thought the experts here might know.

Is it just "the adjective form"? There must be a name for the process that takes a noun and turns it into an adjective.

There's a difference here between the sound and the spelling.

From the noun 'fire' to the adjective we get 'fiery'. From 'wire' we get 'wiry', The pronunciation transformation is regular (the two rhyme).

There are other nouns ending in the sound '-ayr' (or in some varieties '-ah-yer' or '-ahr') which don't form an adjective this way, e.g. tire->tire-like.

Normally one doesn't refer to a suffix change for a noun to an adjective as a conjugation, or a declension (at least in English), or really even an inflection. An adjective suffix is enough. Converting, deriving, or transforming will all work.

I think the general term you are looking for is "derivation". The -y suffix is a "derivational affix" in morphological terms. I'm not aware of a word for the specific process of turning a noun into an adjective.

The fact that the -y adjective formed from "fire" is irregular ("fiery", instead of "firey") is just an accident of the language. There are many other nouns that produce irregular derivations (of all parts of speech): for instance glory + -fy → glorify.

I would call the variations inflections (or inflected forms)

inflection: an ending or other element in a word that indicates its grammatical function in a sentence. (whether it is plural or singular, masculine or feminine, subject or object, and so on)

(Italics mine). I'm sure most people would agree the -y ending is an inflection (often indicating adjectival usage). As Zack says, there's nothing unduly remarkable about the fact that fiery is "irregular" in other respects; I'd still call the ire/ier switch a form of inflection.