Enjoying God Ministries (which I don't know anything about other than the page I will link to) has this summary: The Postmillennial View of the Kingdom of God. They list a number of Biblical quotes which, for completeness, I will add here:
Old Testament: Num. 14:21; Psalms 2:6-9; 22:27-28; 47; 72:8-11; 110:1-2; 138:4-5 (cf. 102:15); Isa. 2:2-4; 9:6-7; 11:6-10; 45:22-25; 65; 66; Jer. 31:31-34; Daniel 2:31-35; Zech. 9:9f.; 13:1; 14:9.
New Testament: Matt. 13:31-33; 28:18-20; John 12:31-32; 16:33; 1 John 2:13-14; 3:8; 4:4,14; 5:4-5; Acts 2:32-36,41; Rom. 11:25-32; 1 Cor. 15:20-26, 57-58; Hebrews 1:8-9,13; 2:5-9; Rev. 2:25-27; 3:7-9; 7:9-10; 11:15; 19:11-21.
One of the more interesting quotes (repeated on monergism (again, I have no other knowledge of the site)) is Psalm 22:27-28:
All the ends of the earth will remember and turn to the LORD, and all the families of the nations will bow down before him, for dominion belongs to the LORD and he rules over the nations.
Personal notes:
The verses I've checked so far, like Psalm 22:27-28, could also be pre-millennialism. I don't see any post-millennialism specific verses. The closest I could find is Matthew 28:19:
Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.
And the second closest is probably John 12:31-32:
Now is the time for judgment on this world; now the prince of this world will be driven out. But I, when I am lifted up from the earth, will draw all men to myself.
To put it more simply: I was a pre-millennialist before I started this study. Now I'm a definite pre-millennialist!
(Another, Psalm 47 appears to be more about military conquest and though it has verses that could be interpreted as millennial, they could just as easily be pre-millennial.)