The show, which first appeared in 2003 at Off-Broadway theatre the Vineyard before transferring to Broadway later that year, ran for five years in the West End after transferring in 2006.
It was previously revived in the capital last May at fringe venue Upstairs at the Gatehouse.
Produced by Sell a Door Theatre Company, Avenue Q will run at Greenwich from 28 April to 11 May 2014 (previews from 23 April), before embarking on a UK tour from 13 May.
Sell a Door's David Hutchinson said: "This is our third major musical production following Spring Awakening in 2011 and Seussical in 2012 and 2013. I have every faith that this parable, which so cleverly pokes fun at the issues young people face going into adulthood, will strike a chord with audiences up and down the country."
Written by Robert Lopez and Jeff Marx, Avenue Q follows recent graduate Princeton through his struggles of living in New York with nothing but a BA in English. He and his neighbours, both people and puppets, struggle through life on Avenue Q - most of them only there because it's all they can afford.
Directed and choreographed by Cressida Carré, the new production is designed by Richard Evans. Casting is still to be announced.
The tour will visit locations including Torquay, Crewe, Manchester, Edinburgh, Blackpool, Sunderland, Croydon and Guildford. For further details visit www.selladoor.com.