The Old Testament didn't mention the kingdom of God nearly as much as the New Testament. However, several things were recognized in the Old testament. One, that Heaven rules. (Dan. 4:26), and more specifically, the King of heaven (Dan. 4:37). The OT also speaks of the God of heaven establishing an everlasting kingdom.
"In the days of those kings the God of heaven will set up a kingdom which will never be destroyed, and that kingdom will not be left for another people; it will crush and put an end to all these kingdoms, but it will itself endure forever. - Dan. 2:44
'Then the sovereignty, the dominion and the greatness of all the kingdoms under the whole heaven will be given to the people of the saints of the Highest One; His kingdom will be an everlasting kingdom, and all the dominions will serve and obey Him.' - Dan. 7:27
Daniel's prophecy I believe is what Jesus brought to man; what Jesus established by His work on the cross. Luke 1:33, in speaking of Jesus says "...His kingdom will have no end." Jesus says in Mark 1:15 that "The time is fulfilled, the kingdom of God is at hand..." fulfilled here I believe is the fulfillment of what Daniel prophecied in the above 2 verses.
"But if I cast out demons by the finger of God, then the kingdom of God has come upon you. - Luke 11:20
In Jesus's teaching He spoke of the kingdom in many different ways. As a man sowing good seeds growing up alongside tares (Matt 13:24-30), as a seed, which starts smaller than the other seeds, but grows to be mightier than the other plants that grow (Matt. 13:31-32), like leaven that grew and leavened 3 pecks (6 gallons, or 42lbs) of flour (Matt. 13:33-35), as a treasure that a man who discovers it would cast off all he has to obtain it (Matt. 13:44), as a pearl which a merchant seeking fine pearls would sell all his inventory to purchase (Matt 13:45-46), as a dragnet cast into the sea, drawing up both good fish, and bad fish (Matt. 13:47-52), a landowner hiring labors to work in His vineyard, some of which did different amounts of labor than others, but all receiving the same reward (Matt. 20:1-16), a king calling people to His wedding feast, first, those whom He had invited, and at their unwillingness (and unworthiness) they were destroyed, then he called those whom His servants found in the highways. (Matt. 22:1-14). And as a mystery:
And He was saying to them, "To you has been given the mystery of the kingdom of God, but those who are outside get everything in parables,
so that WHILE SEEING, THEY MAY SEE AND NOT PERCEIVE, AND WHILE HEARING, THEY MAY HEAR AND NOT UNDERSTAND, OTHERWISE THEY MIGHT RETURN AND BE FORGIVEN." - Mark 4:11-12
"Father, glorify Your name " Then a voice came out of heaven: "I have both glorified it, and will glorify it again."
So the crowd of people who stood by and heard it were saying that it had thundered; others were saying, "An angel has spoken to Him."
Jesus answered and said, "This voice has not come for My sake, but for your sakes. - John 12:28-30
There are over 100 mentions of the Kingdom of God in the New Testiment. The gospel writers later translated Jesus' sermons and parables into Greek. Mark, Luke & John translated Jesus' words as "kingdom of God." while Matthew sometimes used this phrase too, but often he preferred to translate Jesus' Aramaic words as "kingdom of heaven."
It's interesting that the kingdom of God, when compared as a "seed" as mentioned above is at first, unimpressive, but will grow to something magnificent! (Matt. 13:31-32).
The kingdom is also something to be inherited later as well. (Matt. 25:1-46, Heb. 11:1, Rom. 8:23-25)
It is also something to seek first (Matt. 6:33).
for the kingdom of God is not eating and drinking, but righteousness and peace and joy in the Holy Spirit. - Rom. 14:17
but just as it is written,
"THINGS WHICH EYE HAS NOT SEEN AND EAR HAS NOT HEARD,
AND which HAVE NOT ENTERED THE HEART OF MAN,
ALL THAT GOD HAS PREPARED FOR THOSE WHO LOVE HIM." - 1 Cor. 2:9