Q:

Declension is a noun. What is the verb?

Based on Wikipedia article, in linguistics, declension is the inflection of nouns, pronouns, adjectives, and articles to indicate number (at least singular and plural), case (nominative or subjective, genitive or possessive, etc.), and gender.
Declension is a noun. What is a proper verb to say "I want to say different words that appeared from the word day - days, daily etc". How to say it in one word in English?

A:

The verb corresponding to declension is decline. However, your example of day - daily is not an example of declension. It is an example of word formation (with the adverbial suffix -ly). Declension adds/changes inflections, not suffixes.
And here's a little on-topic joke I came across recently:

A verb walks up to a noun in a bar:
--  Hey, babe, wanna go back to my place and conjugate?
-- I decline.

