“In the beginning, God created heaven and earth, which beginning of time, according to this chronology, occurred at the beginning of the night which preceded the 23rd of October in the year 710 of the Julian period.”
In the right margin of the page, Ussher computes the date in “Christian” time as 4004 B.C.
Get the scoop here: The Annals of the World.
I understand he was a meticulous documenter of his calculations…
Do I personally believe God began Creation 6018 years ago tonight?
I’ve no idea. It’s a tingly notion to ponder on this dark, rainy evening. I’m sure I could flex on that by a year or two either way. 
I really must get back to writing Lesson 4 of CLE’s Sunrise Bible 904. For the record, here is the opening paragraph I penned for Lesson 1 of Bible 901:
Let’s go back to the original Zero Hour, that precise point in time when the beginning began. Look again at the timeline. Imagine yourself standing on it, heels pressed against Zero Hour, facing the long expanse of history about to begin. You cannot go back earlier in time, not even one nanosecond. There is no time there for you to step backward into! No time, no history, only . . . an unimaginable, incomprehensible timelessness called eternity, which the Bible sometimes refers to as everlasting:
“The high and lofty One that inhabiteth eternity” (Isaiah 57:15).
“Thou art from everlasting” (Psalm 93:2).
Oh, if you need to read more about his calculations before buying James Ussher’s famous book, you could tackle this:
Bishop James Ussher Sets the Date for Creation.