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"
Best Answer
From the quoted definitions at etymonline, I would suspect that you may be asking the wrong question :)
If I look at the related words in other languages (dag, Tag) for day, it seems the final g has changed into a [j]. The same seems to have happened with (Dutch) leggen -> English lay.
As it is normally pronunciation that defines spelling, and not the other way around, it seems that the i did the job of representing that [j] in daily quite well. However, as a final vowel, the i seems to be uncommon in English, and usually written as a y. (Compare Dutch hooi to English hay).
So where dæglic became daily (the final [k] became [i]), dæg became day. In both cases the g got to be pronounced as [j], but instead of dai, hai or gai, the use of y was preferred to write that sound in a final position.