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"
If I may ...
The origin of the word gold itself can be traced, via German, to the Gothic gulth.
http://www.dictionary.com/browse/gold?s=t
That said, the original Dutch for "guilder" (a golden coin, as well as the official currency of Netherlands until 2002 A.D. is gulden. It was also common in many German territories.
Which would suggest that John Walker may have been wrong when he said
It is much to be regretted that the second sound of this word is grown
much more frequent than the first
in assuming that "the second sound" was a new development rather than the old version that, through a quirk of fate, had regained some of its erstwhile popularity. That is if we wish to trust him at all in this case. He may have thought that the sound had "grown much more frequent" whereas in reality it may have been frequent all along. 18th Century experts are hardly infallible.
Best Answer
At the start of words like gnaw
Yes, at the start of some words like gnaw. Even though the G in gnaw is “silent” in present-day English, it used to represent a consonant sound. So originally, the "GN" in this word wouldn't have been a "digraph" so much as a consonant cluster (albeit a special kind of consonant cluster: the sounds are "tautosyllabic" or in the same syllable). As the spelling suggests, the consonant cluster is thought to have originally been pronounced as [gn], although it may have developed other realizations in certain time periods.
The distinction in pronunciation between words starting with gn- and words starting with n- seems to have been lost sometime in the early Modern English time period (An Introduction to Early Modern English, by Terttu Nevalainen (2006), says the change of gn- to /n/ was "completed in the south in the eighteenth century", p. 128).
In the middle of some compound words like hangnail
There are also some compound words made from native elements that are spelled with -gn-, such as hangnail.
"G" itself isn't very frequent in non-word-initial position in native English vocabulary because of sound changes
The /g/ sound is a bit rare outside of word-initial position in native English vocabulary because it was historically vocalized in many contexts to /j/ or /w/.
The vocalization of G to /j/ (in palatalizing contexts) had already occurred by the time of Old English (at least, in West Saxon dialects), so the letter "G" in Old English spellings could represent the palatal glide /j/ (we have evidence for this from spellings that use G unetymologically to represent /j/ that did not originate from the Proto-Germanic *g sound; e.g. the word for year, cognate to German Jahr, was spelled with the letter G in old English). The Old English word regn that Laurel's answer mentions was probably pronounced something like /rejn/.
The vocalization of G to /w/, which occurred later, seems to have developed via rounding of earlier /ɣ/. An example is the Modern English word owner which the OED says had spellings like agenere, agnere, and ahnere in Old English. The Bosworth-Toller entry is at ágnere.
The pronunciation of "-gn-" as /gn/ is not inconsistent with Latin, or even French origins for a word
The pronunciation of -gn- as /gn/ in many Latinate words is based on spelling (and also partly on certain traditions for pronouncing Latin, which may have themselves have been based on spelling): it doesn't have much if anything to do with Germanic. Note that even though -gn- is pronounced as /ɲ/ most of the time in French words, there are actually some learned French words where /gn/ is used, such as ignition and stagnation.