He replaces David Baker, who has joined Burouj Consulting as UAE country manager.
Moore has been based in Four's Dubai office since late 2006, and was formerly group head of media relations.
Einir Williams, one of the original four founding partners of Four Communications, and head of the agency's brand, property and travel practice, will join him in Dubai.
Williams will spend half her time in Dubai, managing property and travel and tourism clients in the region. The company is also recruiting across all disciplines in Dubai.
Ray Eglington, Four Communications Group's managing director, international, said: "Our operation in Dubai has now been running successfully now for just over two years, growing to offer a full-service marketing communications capability to a wide range of clients.
"But strategically, this is the moment to invest much more in the United Arab Emirates in particular and the GCC in general.
We are bringing more people in to service our new and existing clients, and Greg and Einir have the remit to expand that team further to ensure that we have the best people and infrastructure to be found anywhere in Dubai's communications sector."
Four Communications' clients include property developer Nakheel, Sony Playstation, the Bahrain Economic Development Board and MTC Vodafone.