Many words with the letters que or qui stem from the Latin verb quaerere, which means “ask” or “seek,” and therefore pertain to questions and quests. This post lists and discusses such words.
Quest was originally synonymous with inquest (literally, “search in”), which refers to a legal investigation, but the former word came to apply generally to any search or mission. Now, quest is often associated with chivalric adventures or related journeys in fantasy literature. (Bequest, referring to an act of providing for someone in a will, is the noun form of bequeath and is unrelated.)
Question originally referred to a problem of philosophy or theology but later, by association, pertained to anything intended to prompt an answer or a discussion and came to serve as a verb as well. An act of interrogation is a questioning, someone who questions is a questioner, and an act of a dubious nature (which would prompt observers to question the actor’s morals or motives) is questionable.
Query is synonymous with question as both a noun and a verb. Querent, likewise, is a synonym for questioner but usually in the context of someone who seeks astrological insight; it is rare.
To inquire is to ask, and an act of asking is an inquiry; the latter word is also synonymous with inquest. (The variations enquire and enquiry are associated with British English but are sometimes used by writers in the United States.) Inquisition has the stronger sense of an interrogation; the adjectival form inquisitive implies mere curiosity, but it usually has the connotation of excessive interest. An investigation may also be referred to as a disquisition, although this term may alternatively refer to a long speech.
Request also means “ask” as well as “something asked,” and originally was synonymous with the related verb require, but the latter term came to refer to asking something with the expectation that it must be answered; this imperative sense is matched in the noun form requirement. Something requisite is required in the sense of “necessary,” and a requisition is an instance of asking for something considered essential. The noun and adjective prerequisite, which literally means “required beforehand,” is not to be confused with perquisite (“thing sought”), which is often abbreviated to perk in the sense of “benefits of employment or membership.”
Terms that may not appear to be related but are include the verb acquire (“earn” or “gain,” from the sense “seek to obtain”) and its adjectival forms acquired and acquisitive and noun form acquisition, the verb conquer (“search for”) and its noun form conquest (and the English and Spanish actor nouns conqueror and conquistador), and the adjective exquisite (literally, “carefully sought”).