In the DC Comics version of Greco-Roman mythology there are indeed plenty of stories which pit Ares against the Amazons and those which ally these women with Zeus. There are also some of these comicbook stories in which antagonism occurs between Zeus and the Amazons.
In ancient mythography, however, it does not seem that the Amazons are ever even mentioned alongside Zeus in the same sentence. As for Ares, there are two quite indirect references that serve as a possibility but which I would say still add up to a “No” as well. These are both from the adventures of Theseus.
The War Against Attika [Attica]
The saga of Theseus’ interaction with the Amazons has it that at some point after he became king of Attika, Theseus participated in an expedition to the land of the Amazons either as his own project or as part of the ninth athlos (labour/ task) of Herakles which required that the girdle of the Amazon queen Hippolyta be brought to Argolis. While he was in their territory Theseus abducted either the queen herself or, as is more commonly reported, the queen’s sister Antiope (in one version of the story these two women’s names are swapped around too).
Taking the royal Amazon family member back to Attika with him, Theseus married her and she bore him a son. Some years afterwards Hippolyta (or whatever the Amazon queen’s name was) came to Attika with her entire army and besieged Theseus’ kingdom for three months in a bid to retrieve her sister therefrom.
Theseus’ wife fought by his side against her own sister and her own people, and was run through with a spear by the Amazon Molpadia when Molpadia found out that Antiope wanted to stay with her husband. Molpadia was then killed and the Amazons eventually lost the war, their queen retreating with her remaining forces to Megara and dying there in grief while nursing the mortal wounds she had sustained.
After Theseus’ death a festival, called the Theseia, was instituted to celebrate the great hero king. Prior to this celebration, however, the Athenians always performed a sacrifice in honour of the fallen Amazons. The place by the Areiopagos, the "Crag/Hill of Ares," at which the warrior women had established their camp was named the Amazonion [Amazonium], and the citadel and lofty towers which the women had built upon it rivalled those around Theseus' palace.
In his Life of Theseus, Plutarch says that prior to his first foray into combat against the Amazons, an oracle had instructed Theseus to offer a sacrifice to Phobos, the daimon son of Ares who personified fear, especially the fear which leads to retreat from the battlefield.
According to Pausanias’ Description of Greece, at the city of Troizenos [Troezen], there was, on the way down to the harbour, a place called Kelenderis. A particular spot here had been named Genethlion, “Birthplace,” because it is where Theseus was born. Also on this spot Theseus had waged war against the Amazons during their attack upon his domain.
Before this place [Genethlion] is a temple of Ares, for here also did
Theseus conquer the Amazons in battle.
Pausanias does not explicitly say that it is the case, but Florence Mary Bennett, in her book Religious Cults Associated with the Amazons (p. 58), interprets this to mean that Theseus is the one who built the Ares temple at Genethlion in order to commemorate his victory over these women.
The implication here is that Ares favoured Theseus over the Amazons. That, however, need not be the case. After all, the Amazons’ enemies the Athenians themselves performed religious ceremonies in honour of the Amazons that they themselves had killed. Granted that the Athenians were notorious down into later times for being hyper-religious, trying to be careful to honour every last deity and divinity that they came across. This was so much so that by the Roman period it would be remarked that it was easier to find a god (i.e. an idol) than a man in Athens.
During the trial of Orestes on the Athenian Areipoagos in Aeschylus’ Eumenides, the goddess Athena points out to the court that the Amazons performed sacrifice to Ares on this very spot during their siege of Attika a generation prior to this trial. That the women's sacrifices did not avail them of victory might strengthen the notion that Ares was in some way opposed to the Amazons in this instance. It is, however, a glaring phenomenon in the mythology that Ares, even when he himself participates in wars personally (maybe especially when he does), is really not good at winning them, in spite of his immense size! In his most dramatic battlefield altercations on Earth, those against Herakles and Diomedes, he is so badly wounded by these mortals that he must repair to the home of the gods for an urgent patch-up in the divine infirmary.
An additional consideration to be made regarding the war between Theseus and the Amazons is that one of Ares’ daughters, Theseus’ wife (typically Antiope, although Plutarch notes that Cleidemus names her Hippolyta, who nonetheless by all accounts is also Ares’ offspring), is fighting on the side of the Athenians. If we are to understand Theseus’ ceremonial activity in connection with this conflict (the sacrifice to Phobos and the construction of the Genethlion shrine to Ares) as an indication of Ares’ disfavour towards the Amazons, it could conversely be interpreted that the one Amazon who is fighting against her own people might be the one bearing her father’s favour in this case.
The characterisation of the DC Amazons as a society of peace-loving yet formidable warriors can be rendered as an unintentional exploration of the other side of this nation’s origin in the ancient myths. Admittedly this theory is based entirely on the etymology of one character’s name, which is by no means overtly consequential in any story. As your Argonautika quote says, the first Amazons were the daughters of Ares by an obscure nymph named Harmonia in the “Wood of Akmon”, which I find to be a most tantalising reference indeed.
DC Comics and the 2017 Wonder Woman movie may have ended up inadvertently exploring the maternal origins of the Amazons in a deeper understanding than the ancient Greco-Roman writers. It’s as if someone had asked these questions: What if the name of the Amazons’ mother Harmonia, “Harmony,” is an indication that she is a personification of peace and tranquillity just as much as Ares personifies senseless violence and en masse belligerence?
And if so, what if the Amazons had taken after their mother rather than their father? This line of theory breaks down eventually because in the DC lore Ares is not the Amazons’ father nor in fact is Harmonia their mother. But it points to an interesting nuance that could conceivably have stronger roots in the original version of this mythology.
The references in Plutarch and Pausanias would appear to be the closest one could get to a conflict between Ares and the Amazons, and, as stated before, I don’t think that they can be counted as such in any meaningful sense.
Beyond him siring the Amazons upon the nymph Harmonia, him siring a few more daughters upon Otrere and Oreithyia, and him giving Hippolyta her famous girdle, Ares really has no direct interaction with any Amazon in the ancient mythology, whether positive or negative. If anything Ares' relations with the Amazons is made conspicuous by his absence among them!
In their two most famous confrontations before the Trojan War, their fights with Herakles and with Theseus, Ares doesn't show up at all like he does during Herakles' duel against Ares' son Kyknos [Cycnus] or during Herakles' attack on the city of Pylos. Moreover the battle with Herakles is the result of Ares's mother Hera herself making an appearance for the purpose of provoking violence between Herakles and the warrior women. Beyond this apparent neglect on Ares's part, the Amazons are certainly, to go back again to Apollonius Rhodius, depicted as bellicose and pugnacious, which should make them just Ares' kinda gals!
Beyond Ares' overall MIA status among the warrior women in question, the Amazons, as a matter of fact, barely have contact with any deity in general. There are only two exceptions to this that I am aware of: Artemis and Dionysos [Dionysus].
In Diodorus Siculus’ Bibliotheka Historika, the Amazons Kelaino [Celaeno], Eurybia and Phoibe [Phoebe] are said to have been hunting companions of Artemis. They were slain, all three of them, fighting to defend their queendom when Herakles came to fetch Queen Hippolyta’s girdle.
Dionysos, during his tumultuous campaign around the world, made war on the Amazons. Pausanias says that some of them fled before him, ending up as suppliants at the temple of Artemis Ephesia (Artemis of Ephesos [Ephesus]), which sanctuary a queen of theirs had built long before then. As he was in the habit of doing, such as when he had conquered the Indians, he added some of his defeated foes to his retinue. “Those Thermodontian hordes” of “warrior women”, Seneca tell us, were turned into Mainades [Maenades].
Dionysos' genealogy is intriguing background character information which nevertheless does not seem to affect the story: Dionysos himself is Ares' offspring, since Ares is his great-grandfather. Just like in the encounters with Herakles and Theseus, here is another major conflict in which the Amazons are engaged but in which Ares does not feature at all, although nothing in what we know about the story indicates that it is because he was supporting his more distant descendant, the god of wine, over against his daughters.
Dionysos bears a connection with Harmonia as well: Dionysos' mother Semele is the daughter of Harmonia daughter of Ares. This Harmonia, however, is explicitly said to have been the goddess of harmony and concord; she is Ares' daughter rather than his lover; and she becomes queen of Thebes after marrying the city's founder Kadmos [Cadmus], to whom she bears Semele. The coincidence that there are two Harmoniai (Harmonias) who are both associated with Ares is fascinatingly curious. Aaron Atsma's analysis of it is that:
The Amazonian Harmonia does not appear to have been identified with
the Theban Harmonia. They both have the Greek word harmonia,
"harmony," for a name but in the case of the Amazon it no doubt
referred to the harmony of military formation whereas the Theban queen
was harmony in the sense of marital bliss.