What is the reason behind this? Do the noun and the verb have different derivations?
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First, notice that many (possibly even most) English speakers use "indexes" as the plural. But let's ignore that, because many people do say "indices".1
The Latin noun "index" is third-declension, so its plural is "indices" in both nominative and accusative case. The English noun "index" sounds almost identical, it's spelled exactly the same, and its meaning is pretty close.
More importantly, there's a semi-regular rule in English that Latinate "-x" nouns often keep their Latin "-ices" plurals.2, 3 So, we don't need to say anything implausible, like that English borrowed both "index" and "indices" separately, or that anyone who says "indices" is applying Latin declension.
Meanwhile, there is no Latin verb "index" to be borrowed. There is a first-conjugation verb "indicare", of course, and the Latin noun "index" was derived from the verb's first singular present "indico", but it wouldn't conjugate to "indices". And at any rate, we don't have the verb "indicare"; we say "to index", "I index", "I indexed", not "to indicare", "I indico", "I indicavi". So of course we also say "he indexes", not "he indicat".
In fact, the English verb "index" was formed by verbing the already-borrowed noun, not by borrowing a verb. It seems to be pretty recent—dictionaries from a century ago (e.g., the 1911 Century Dictionary call it "rare" or "nonstandard" or "American".
English is full of verbs created out of nouns like this—and the verb's conjugation isn't affected by whether the noun is irregular. So, "index" is conjugated like any other English verb.
1. Looking at a handful of dictionaries, they all list both forms. Most list "indexes" first, but nobody says "indices" is obsolete or variant or anything like that.
2. One good way to spot a semi-regular rule is to look for people overgeneralizing it. And there are plenty of examples. For example, people (young learners, insecure hypercorrectors, literate word-players making a joke, …) sometimes say "prefices". It's common enough that it's even listed as a nonstandard plural in many dictionaries. The noun "prefix" is borrowed from French, and the closest Latin is "praefixus" which pluralizes to "praefixi", it does end in "-x" and certainly sounds Latinate, so people apply the rule inappropriately.
3. You can get into theoretical arguments about whether there's a morphosyntactic rule that turns "index" + "-s" into "indices" on the fly, a rule "in the lexicon" that forms the word "indices" out of "index", or just a word "indices" that's part of an semi-regular pattern that extends by analogy, but that isn't really important.
The Latin noun 'index' has plural 'indices' (see https://www.online-latin-dictionary.com/latin-dictionary-flexion.php?parola=index). For English usage of noun 'index' and verb 'to index', see this SE thread: https://english.stackexchange.com/questions/61080