She will join below-the-line advertising agency Eyes Down Media next month in the newly created position of print director.
The agency, which is based in London, handles in-store advertising for supermarket Morrisons.
"Morrisons' takeover of Safeway means they will need to expand their range of services and I bring that side to it," said Graham.
"EDM is also developing a contract publishing division, handling everything from small projects to magazines," she added.
Sales and marketing director Ninian MacGregor said: "Fiona brings an enormous amount of expertise on the production side. There are a lot of places we hope to go with her."
Graham said her membership of the Institute of Direct Marketing gave "a broad range" to her production capabilities. "I've worn an awful lot of hats here but I've bothered to get qualified too," she said.
Spectator production controller Alida Campbell will step up to become production manager on Graham's departure. The magazine is part of the Telegraph Group, whose future ownership is subject to an ongoing bidding process.