How did the wise men (Magi) know the country where the new king was born?

Being a king of Israel, it is a responsibility of Herod to know the scriptures and keep it. As what is written in II Chronicles:

34:31 And the king stood in his place, and made a covenant before the LORD, to walk after the LORD, and to keep his commandments, and his testimonies, and his statutes, with all his heart, and with all his soul, to perform the words of the covenant which are written in this book.

It is but natural for the wise men to come to Herod because he is the king that time. It is called courtesy.

@Mawia remember, they asked the question. You cannot get any information from any other books except from the scriptures.

They knew the country because it was phophesied by Micah, In Micah 5:2 it reads:

But thou, Bethlehem Ephratah, though thou be little among the thousands of Judah, yet out of thee shall he come forth unto me that is to be ruler in Israel; whose goings forth have been from of old, from everlasting.

A new brand of liberal theologians are trying to convince everyone that pagan mystics using some kind of black magic or divination discovered the secret of the “new” star, while the Jewish religious leaders were left in the dark (even if the priests and Pharisees were corrupt and blind to spiritual things, God always leaves Himself a remnant; godly people like Simeon, Anna and others were expecting the Messiah to come at that time - see Luke 5:25-38).

The Magi’s most famous and God-honoring predecessor was Daniel, who, along with his three Hebrew friends, were taken from Jerusalem to Babylon, hundreds of years earlier, to serve in Nebuchadnezzar’s court.

In the third year of the reign of Jehoiakim king of Judah came Nebuchadnezzar king of Babylon unto Jerusalem, and besieged it. And the king spake unto Ashpenaz the master of his eunuchs, that he should bring certain of the children of Israel, ... Now among these were of the children of Judah, Daniel, Hananiah, Mishael, and Azariah:
[Daniel 1:1,3,4,6]

One of Daniel’s most fascinating prophecies came during the interpretation of the king’s dream about the huge statue with various parts of gold, silver, brass, iron and iron mixed with clay:

There is a God in heaven that revealeth secrets, and maketh known to the king Nebuchadnezzar what shall be in the latter days. And in the days of these kings shall the God of heaven set up a kingdom, which shall never be destroyed: and the kingdom shall not be left to other people, but it shall break in pieces and consume all these kingdoms, and it shall stand for ever. Forasmuch as thou sawest that the stone was cut out of the mountain without hands, and that it brake in pieces the iron, the brass, the clay, the silver, and the gold; the great God hath made known to the king what shall come to pass hereafter: and the dream is certain, and the interpretation thereof sure. [Daniel 2:28, 44, 45]

We see that God is going to set up an eternal kingdom, but when is that going to be? Daniel gives them the answer in this next prophecy:

Seventy weeks [of years] are determined upon thy people and upon thy holy city, to finish the transgression, and to make an end of sins, and to make reconciliation for iniquity, and to bring in everlasting righteousness, and to seal up the vision and prophecy, and to anoint the most Holy. Know therefore and understand, that from the going forth of the commandment to restore and to build Jerusalem unto the Messiah the Prince shall be seven weeks, and threescore and two weeks: the street shall be built again, and the wall, even in troublous times. [Daniel 9:24,25]

There is a lot of heavy prophecy in these few verses, but for now we will just pick out a couple that are relevant for this question. Seventy weeks are determined upon the people of Israel. There will be sixty-nine weeks [of years] or 483 years from the time the command is given to rebuild Jerusalem until the time of Christ’s arrival. Unfortunately Messiah will be cut off, and the final seventieth week is still in the future. This will be seven years of desolations (tribulations) before He sets up the eternal kingdom.

But will there be some special sign that the Messiah has come? How would the Magi know where to look for the new King? Many scholars believe that Baalam’s prophecy from the book of Numbers chapter 24 is the basis of the, God-fearing, wise men’s quest:

I shall see him, but not now: I shall behold him, but not nigh: there shall come a Star out of Jacob, and a Sceptre (sovereignty) shall rise out of Israel, and shall smite the corners of Moab, and destroy all the children of Sheth. And Edom shall be a possession, Seir also shall be a possession for his enemies; and Israel shall do valiantly. Out of Jacob shall come he that shall have dominion, and shall destroy him that remaineth of the city. [Numbers 24:17-19]

There shall come a star out of Jacob and a sceptre out of Israel. And out of Jacob shall come the Messiah. He is the One who will, ultimately, usher in God’s eternal Kingdom.

The wise men were looking for a "Star out of Jacob (Israel)," 483 years after the command to restore and rebuild Jerusalem.

According to Dr Michael Molnar, the most likely explanation is that the Magi, being astrologers, used astrology to identify the birth of a king in a certain country, because there was an astrological conjunction of (most) of the Classical planets in a given spot of the sky. That spot was in the Zodiacal constellation of Aries, the Ram, and according to 1st Century Babylonian astrology, the Ram was associated with Judaea. So they interpreted that conjunction as being a sign that a great king was to be born in that Western country.

Note that I'm not a particularly liberal kind of Christian, and I'm not arguing that astrology somehow works; but assuming that the pagan Magi did not have a direct revelation from the God of Abraham, Isaac and Jacob, they must have had a sign that they could reasonably interpret to be the birth of a king in Judaea. And so far, that seems to be the most likely theory if we only consider what the Magi knew — or rather, what we currently know about what they knew.

I think what you're looking for is detailed in Matthew 2:1-12:

After Jesus was born in Bethlehem in Judea, during the time of King Herod, Magi from the east came to Jerusalem and asked, ‘Where is the one who has been born king of the Jews? We saw his star when it rose and have come to worship him.’

When King Herod heard this he was disturbed, and all Jerusalem with him. When he had called together all the people’s chief priests and teachers of the law, he asked them where the Messiah was to be born. ‘In Bethlehem in Judea,’ they replied, ‘for this is what the prophet has written:

‘“But you, Bethlehem, in the land of Judah, are by no means least among the rulers of Judah; for out of you will come a ruler who will shepherd my people Israel.”’

Then Herod called the Magi secretly and found out from them the exact time the star had appeared. He sent them to Bethlehem and said, ‘Go and search carefully for the child. As soon as you find him, report to me, so that I too may go and worship him.’

After they had heard the king, they went on their way, and the star they had seen when it rose went ahead of them until it stopped over the place where the child was. When they saw the star, they were overjoyed. On coming to the house, they saw the child with his mother Mary, and they bowed down and worshipped him. Then they opened their treasures and presented him with gifts of gold, frankincense and myrrh. And having been warned in a dream not to go back to Herod, they returned to their country by another route.

So, they headed first West (because of the sign of the star), then to the capital of the country they found themselves (Jerusalem), then zoomed in further by heading for Bethlehem (on the advice of Herod's staff) and finally to the exact house, because the star was above it.

_{(Perhaps they should simply have followed the star all the way, instead of detouring via Jerusalem?)}