ANSWER: In order to figure out what events, wars or fulfilled prophecy between the testaments occurred we should first know some approximate dates. Malachi, the last book in the Old Testament, was written roughly around 400 B.C. The conception of John the Baptist, the first major event listed in the New Testament (Luke 1), took place in the middle of 6 B.C. (see our timeline of Jesus' birth for more info).
What wars occurred during the period roughly between the Old and New Testaments (400 B.C. to 6 B.C.)? A key military event, famous even today, took place between Alexander the Great and the Persian Empire. Both parties went to war with Alexander defeating the Persians, 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).
Another war that took place during this time period was between the Seleucid kingdom and the Egyptians led by the Ptolemies. Perhaps the most famous person of the Ptolemy line to rule Egypt in this period was Cleopatra, who committed suicide in 30 B.C. The Seleucids and the Egyptians fought back and forth over Palestine, Judea and the Holy Land. The land of Palestine ultimately fell to the Hellenistic Seleucid Empire, ruled by Antiochus IV Epiphanes, in about 175 B.C.
The Jews 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 and 11:31, and is covered in the books of the Maccabees commonly found in Catholic Bibles.
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. According to tradition it miraculously lasted eight days. Please see our article on Hanukkah for more information. The Jews had effective independence until Pompey, the Roman politician and general, invaded the Holy Land in 63 B.C.
There are other wars and events, 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. with the Athenian empire ultimately being defeated.
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 and events in China, which aren't mentioned evenly obliquely in Scripture, during this time. They ultimately resulted in the first Chinese Emperor coming to power in 221 B.C.