What is kingdom of God to the Hebrews?

It should be the same as Christians look forward to. Unfortunately it is not. The Jewish people did not see the ruler (Savior & Messiah) being Jesus. Some Christians do not see it as a place on this Earth.

Abraham was told of the Kingdom in Genesis 12:7. The land will be given to him and his 'seed' forever. ('seed', see Galatians 3:16). David knew that a King would rule righteously from his very throne and that kingdom would never end. Daniel interpreted the destruction of man's kingdoms and the replacement with the Kingdom of God.

"The Earth shall be full of the knowledge of the glory of the LORD, as the waters cover the sea" Hab 2:14

A small bit from Isaiah 11

"6The wolf also shall dwell with the lamb, and the leopard shall lie down with the kid; and the calf and the young lion and the fatling together; and a little child shall lead them."

Also see 1 Corinthians 15

Again, the Hebrews knew of this Kingdom and the promises made to Abraham, Isaac, and Jacob but did not know the Christ when He came first, but they will when He returns. They did not see that the Elijah that would announce the Messiah was John the Baptist.