To become naturally intelligent, turkeys would need to go along many of the same pathways as other intelligent species on Earth.
Hunting or being omnivorous would seem to be a good fist step, since they not only can access more and different sources of food than the competition, but also would need to develop some intelligence thunk prey and find and exploit different food sources.
Most intelligent animals on Earth also are social, living and working together in groupings like troops, packs and pods. Being social and cooperative seems to be important in developing sophisticated communications between members which leads to language. A flock of photo intelligent turkeys would resemble a troop of baboons, perhaps, more than a standard flock of birds.
As for morphology, the turkey will loose a lot of its bird like features and develop in ways which support both the new lifestyle and a brain large enough to support intelligence. The "face" will become more like that of a raptor with forward facing eyes and other adaptations to support hunting (a serrated beak, perhaps) or omnivorous behaviour (a beak sharp enough to cut chunks of meat from prey but also strong enough to crack nuts or seed pods).
This evolution was simulated with a species of dinosaur called the Troodon, which would already have been much smarter than a turkey to start with. A hypothetical intelligent descendant of the Troodon is illustrated below:
We now know that most dinosaurs were covered in protofeathers, and the Troodon was theropod (the ancestor of birds), so would have likely retained this feature for warmth and sexual display. It is also quite possible that the evolved turkey would be more bird like than this thought experiment, for example retaining a longer tail for balance but having wings become vestigial and manipulating objects with the feet instead.
Reconstruction of a Troodon. An intelligent Troodon might have a larger head and highly developed feet
A highly intelligent turkey would not be very appealing to the human eye, and the Ancestors would certainly have either tried to domesticate them in the distant past (like they did with wolves as the ancestors to dogs), or hunted them to extinction as an existential threat.