The 24-year-old agreed a three-year deal at Edgeley Park yesterday after a £100,000 fee was agreed with his club Linfield.
In his time with the Northern Irish double-winners Thompson scored 152 goals in 234 appearances.
"I know it's going to take me time to adjust and get in to the swing of things but I'm confident once that happens I'll do well," said Thompson. "I've always scored goals whatever level I've played at.
"They may not always have been spectacular goals but a goal's a goal. That would be a big thing I'd like to bring to the team - putting the ball in the back of the net."
County had to beat off competition from Norwich City to sign Thompson, who has won seven caps for Northern Ireland.
He added: "From what I know County are a good family club and I'm looking forward to going places with them.
"This move gives me an opportunity to play on a better platform. To be honest it will be a pretty big step up in standard."
Thompson was not due to feature in today's pre-season friendly at Cammell Laird.
He will return from Belfast before Tuesday's friendly at Rochdale, but is more likely to make his first appearance in a County shirt against a Liverpool XI at Edgeley Park next Saturday.
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