Why is "conquer" pronounced /'kɔŋkɚ/ and not /'kɔŋkwɚ/?

In English "qu" is always used as a digraph.

The letters "que" represent the sound [k] at the ending of many words: unique, technique, antique, physique, clique, grotesque.

However, the combination "qu" has the sound [kw] instead of [k] at the beginning and in the middle of many English words: queen, quarter, quite, quiet, question, squad, squirrel, require, inquire.

Then, why is the "qu" in conquer pronounced [k]?

Would it be c+on+qu+er or c+on+que+r?

Is it anything to do with its Middle English form?

Origin of "conquer":

1200–50; Middle English conqueren < Anglo-French conquerir, Old French conquerre < Vulgar Latin *conquērere to acquire (for Latin conquīrere to seek out). (Dictionary.com Unabridged)

Great question! It seems that this word was originally pronounced with /kw/, as in "quiet," but the /w/ ended up being lost because the following vowel was unstressed.

As Andrew Leach says, in Modern French <qu> generally represents a single consonant /k/, due to a sound change from /kw/ to /k/ that occurred at some point. However, the word conquer was not taken from Modern French: it was taken from "Anglo-French," as you say, and in this dialect of French "qu" does seem to have generally been pronounced /kw/:

words like quit, question, quarter, etc, were pronounced with the familiar “kw” sound in Anglo-Norman (and, subsequently, English) rather than the “k” sound of Parisian French. ("Norman Conquest" – The History of English, Luke Mastin)

The words exchequer, chequer, lacquer, liquor are graphically similar, but they're different from conquer in the specifics of their etymology, so unfortunately a comparison with the pronunciations of these words is not conclusive. According to the Oxford English Dictionary (OED):

• chequer and exchequer are ultimately derived from late Latin scaccārium, so the <qu> in these words just reflects the Parisian convention of writing /k/ as <qu>. I can find no evidence that it was ever pronounced as /kw/ in any variety of French or English. In older forms of French, it was apparently spelled with a <k>, and in English, spellings with <k> or <ck> are attested earlier than spellings with <qu>.

• lacquer is derived from "obsolete French lacre (17th cent.)." The spelling with <qu> is again due to Parisian spelling conventions; in particular, the OED suggests the French word laque "lac" influenced the spelling.

• liquor does in fact have some etymological basis for the <qu>: it ultimately comes from Latin liquor, related to the root seen in words like liquid. However, the /kw/ in this word seems to have been simplified to /k/ fairly early on (possibly the process was quickened by the roundedness of the following vowel). We find <c> and <k> used in some of the earliest attested English spellings in the OED. A similar case spelled with <gu> seems to be the word "languor", which is ultimately from Latin languor, but which had the following spellings among others in Middle English: langre, langur, langore, langoure.

So essentially, for all of the preceding words, Andrew Leach's answer seems to be the best explanation for why they are spelled with <qu>: it's just a French-influenced spelling of the sound /k/.

For conquer, on the other hand, the "Anglo-French" etymology does suggest the pronunciation should have a /kw/. And there does seem to be some evidence that it was pronounced with /kw/ at the time it entered English. The OED lists some Middle-English spelling variants that use the letter w, cuncweari and conqwere. It also seems like there weren't any spellings in Middle English with just <c> or <k>, which we might expect to see if the word was pronounced without an /w/ in this time period.

As Peter Shor mentions, there don't seem to be any words in present-day English that end in /⁠-⁠kwər/ (or, for that matter, /⁠-⁠wər/ preceded by any other consonant). Janus Bahs Jacquet made the point that this might be considered a phonotactical restriction; I thank him for pointing this out. A Google search turned up the following passage by Julia Schlüter from "Early Modern English: Phonology" (in Historical Linguistics of English: An International Handbook, edited by Alex Bergs and Laurel Brinton):

A minor consonantal change limited to a certain number of lexemes and dating to Early Modern English is the disappearance of /w/ when following another consonant and preceding a rounded back vowel, e.g. in sword, two and who, and somewhat more systematically in unstressed syllables, e.g. Southwark, conquer, answer. In some further items, e.g. swollen, swoon, swore, awkward, boatswain, forward, housewife, and pennyworth, /w/ was later restored on the basis of the spelling or of related words.

As this passage says, there are a number of exceptions to this sound change.

For example, Peter Shor pointed out that this kind of simplification does not seem to have taken place (or it was reversed) in the related word conquest. The primary stress in conquest, at least in modern pronunciations, is on the first syllable, but the /kw/ comes before a non-reduced vowel /ɛ/ rather than the schwa. Also, there is a similar word quest where the /kw/ does come at the start of a stressed syllable; this may have contributed by analogy to the maintenance or restoration of /kw/ in conquest.

Other words where post-consonantal /w/ before an unstressed syllable was retained, re-introduced, or just plain introduced in unstressed syllables include relinquish (ultimately from classical Latin relinquĕre, via French relinquiss-); extinguish, distinguish (ultimately from Latin words ending in -tinguĕre); anguish (from Old French anguisse, angoisse < Latin angustia < anguĕre, a variant of the verb angĕre); languish < Anglo-Norman and Middle French languiss- < ultimately from classical Latin languēre); vanquish (complicated; the OED says “< Old French vencus past participle and venquis past tense of veintre ( < Latin vincĕre), modern French vaincre to conquer, overcome; the ending was finally assimilated to that of verbs from French stems in -iss-”; to me it looks like the /w/ must also be due to analogy at some point since Latin, since it is not present in any form of Latin vincĕre).

The place names mentioned in the following question may be other examples of this sound change: "The mysterious, unenunciated "w" in the "-wich" of English place names"

Peter Shor's examples all come from French. I believe you almost answered your own question.

That is, the words had a /k/ sound before they were assimilated into English.

Because English is a silly and irregular language :) Peter Shor's examples in his comments to you point out that this is actually a fairly common irregularity, that is to say there is no guarantee as to how the letters "qu" will be pronounced.

From Consonant Clusters 29 qu- /kw/ (Advanced)[1]

Although now written with a consonant and a vowel as qu-, the /kw/ sound is made up of two consonants so falls under the consonant cluster category. The letter q is used the most rarely of all English letters with the exception of z. In the Old English period it was very rarely seen, as even the sound /kw/ was usually represented by the letters cw- as in cwen "queen". However, in the Middle English period, English began using q more frequently due to the massive influence of French, which used it much more. Q has thus taken on two sounds /kw/ and /k/, the latter often being spelt qu- in French as in words like plaque /plæk/, clique /kli:k/ and cheque /tʃek/, which English has borrowed from that language.