I think the "kingdom of heaven" refers specifically to the millenial kingdom, while "kingdom of God" refers to God's rule during all ages.
This is much easier to disprove, than prove.
To disprove it, just find an example where Matthew used "kingdom of heaven" to refer to a time which is not the millennium.
As far as I know, he never did.
Below are examples where Matthew referred to a part God's kingdom which is not the millennium.
In referring to these times, Matthew used the term, Kingdom of God, rather than his usual "Kingdom of the heavens".
Mat 12:28 referring to when Jesus was on earth. :
28 But if I by the Spirit of God cast out demons, then indeed the kingdom of God is come upon you.
Mat 21:43 referring to the Old Testament time. :
43 Therefore I say to you, that the kingdom of God shall be taken from you and shall be given to a nation producing the fruits of it.
Matt 16:28 referring to Pentecost. (Compare to Mark 9:1; Luk 9:27)
Mat 16:28 But I say unto you of a truth, There are some of those standing here who shall not taste death until they shall have seen the kingdom of God.
Mark 9:1 And he said to them, Verily I say unto you, There are some of those standing here that shall not taste death until they shall have seen the kingdom of God come in power.
Luk 9:27 But I say unto you of a truth, There are some of those standing here who shall not taste death until they shall have seen the kingdom of God.
For reference to the above, the church age is also called the Kingdom of God:
Rom 14:17 17 for the kingdom of God is not eating and drinking, but righteousness, and peace, and joy in the Holy Spirit.
1Cor 4:19 but I will come quickly to you, if the Lord will; and I will know, not the word of those that are puffed up, but the power.
20 For the kingdom of God is not in word, but in power.
Why did only Matthew pick up on this term? Maybe because He was more knowledgeable of the Old Testament and Hebrew than the other 3, as indicated by his more extensive quotes of the OT.
Am curious and would like to know if there are any Biblical passages which relates Kingdom of Heaven/God with Millennial Kingdom? – @Ashish Kumar
Great question. I'll give you some verses where the kingdom of heaven refers to the millennium. It is not a proof because a similar passage may also be in Luke, where Luke calls it "Kingdom of God". This is because the millennial kingdom is also part of the kingdom of God, so it can also be called the kingdom of God.
A stronger evidence, I think, is that whenever Matthew referred to a part of the Kingdom of God which is not the millennium, Matthew used the term, "Kingdom of God" rather than his more-often-used term, "Kingdom of Heaven". Those are the preceding verses.
The Lord likened the kingdom of the heavens to a king making a wedding feast for his son. There is a similar parable in Luke 14:15.
Matt 22: 2 The kingdom of the heavens has become like a king who made a wedding feast for his son,
Luke 14:15 And one of those that were lying at table with them, hearing these things, said to him, Blessed is he who shall eat bread in the kingdom of God.
16 And he said to him, A certain man made a great supper and invited many.
Revelation says that the millennial kingdom is a wedding feast:
Rev 19: 7 Let us rejoice and exult, and give him glory; for the marriage of the Lamb is come, and his wife has made herself ready.
8 And it was given to her that she should be clothed in fine linen, bright and pure; for the fine linen is the righteousnesses of the saints.
9 And he says to me, Write, Blessed are they who are called to the supper of the marriage of the Lamb. And he says to me, These are the true words of God.
Also, Mat 25 likens the kingdom of the heavens to virgins going out to meet their bridegroom:
Matthew 25:1 Then shall the kingdom of the heavens be made like to ten virgins that having taken their torches, went forth to meet the bridegroom.
After the millennium, the Holy City comes down from heaven to earth to remain on the earth. So where was the Holy City during the millennium? It was in heaven. So it is called the kingdom of heaven during the millenium.
Rev 21:2 And I saw the holy city, new Jerusalem, coming down out of the heaven from God, prepared as a bride adorned for her husband.
3 And I heard a loud voice out of the heaven, saying, Behold, the tabernacle of God is with men, and he shall tabernacle with them, and they shall be his people, and God himself shall be with them, their God.