ANSWER: In order to figure out what wars or fulfilled prophecy occurred during this period, we should first know the approximate dates for it. Malachi, the last book in the Old Testament, was most likely written roughly around 400 B.C. A reasonable date for the beginning of John the Baptist's ministry, considered the beginning of the New Testament period, is the early part of 26 A.D. (shortly before Jesus' ministry began in the Fall).
So then, what wars occurred during the period roughly between the Old and New Testaments? A key war, famous even today, broke out between Alexander the Great and the Persian Empire, during which he defeated them in several major battles from 333 to 331 B.C.
After Alexander died in 323 B.C. various wars broke out among his generals for control of his empire, which was eventually split it into four parts. These wars and their outcome were predicted by Daniel (see Daniel 11:3 - 4; 8:3 - 8, 20 - 22). Daniel 11 is largely about the continuing conflicts between the Seleucid and Ptolemaic kingdoms (verses 5-39), which were ruled by successors of Alexander's generals. The King of the North is the Seleucid kingdom, which roughly corresponds to Iran, Iraq, and Syria today. The King of the South refers mainly to Ptolemies, who ruled over Egypt. (Cleopatra, so famous for her dealings with the later Roman generals and politicians Julius Caesar and Mark Anthony in the first century B.C., was the last of these Greek rulers of Egypt). These two kingdoms fought back and forth over Palestine, Judea and the Holy Land. The land of Palestine fell under the Hellenistic Seleucid Empire, ruled by Antiochus IV Epiphanes, in about 175 B.C.
The Jews even today commemorate at Hanukkah the result of wars that gave them at least temporarily effective national independence against the Seleucid. This war was is also mentioned in Daniel 8:9 - 14; 11:31, and is covered in the two books of the Maccabees. (These two books are found in the Old Testament in Catholic Bibles and are part of the Apocrypha, which Protestants and Jews don't recognize as part of the canon of Scripture but Catholics do).
The Seleucid Greek ruler, Antiochus IV Epiphanes, persecuted the Jews very harshly for their religion, including ordering the burning of all the copies of the Old Testament. He also prohibited circumcision and outlawed the animal sacrifices. He even offered pigs up at the Temple of Jehovah, and had it made the Temple of Zeus instead!
Rather astonishingly, and one has to see this as an outcome that God desired, the Jews successfully revolted against the Seleucid during the period 167-164 B.C. Although it's a dubious story historically, Jews traditionally say that when the temple was rededicated in 164 B.C. that only one vial of undefiled holy oil could be found. It was supposed to last for only one day, but it miraculously lasted eight days. The event of the oil lasting eight days is celebrated annually by Jews during Hanukkah. The Jews had effective independence until Pompey, the Roman politician and general, invaded the Holy Land in 63 B.C., where it then became a Roman province.
There are other wars, of course, that occurred during this period that aren't mentioned in the Bible.
Near or just after the close of the Old Testament a major war broke out between the Greek city states, with Athens on one side and Sparta on the other, called the Peloponnesian War. It lasted from 431 to 404 B.C. It was a war Athens lost. Thucydides recounts most of this war famously, in what is perhaps the first reasonably objective work of history. The Roman Republic's great external expansion also occurred during this time (including its conquest of Carthage during the three Punic wars), and its conversion over to an empire under the first Roman Emperor Augustus (reigned 31 B.C. to 14 A.D.). This prepared the empire for the birth of Christ and the start of the New Testament period. Augustus took over Rome after an terrible period of civil war and political conflict within the Roman Republic that had lasted, off and on, for decades. There also were major wars in China, which aren't mentioned evenly obliquely in Scripture, during this time, which ultimately resulted in the first Chinese Emperor taking over in 221 B.C.