That's kind of a two part answer. A turkey is actually old enough to breed at about 9 months, however, turkeys have a definable 'Breeding season' which usually starts about February or March. So, my birds will actually be old enough to breed and lay eggs in November but they probably wouldn't until they're a year old.
Domesticated turkeys lay in the spring and the fall - The wilds usually only in the spring- if they are far enough south sometimes they can have 2 clutches a year.
The amount of eggs layed depends on if you collect the eggs or not. If you let them go in the nest the hen will lay 20-ish and then sit on them. If you collect them every day she will lay many many more. The spring lay can go on for a couple months with the hens laying an egg a day or every other day. The fall lay is shorter.
Here in NC our climate is pretty mild so ours start laying early. The Beltsvilles started laying late Dec and were in full lay by Jan, The Hollands and Midgets were next - mid to late Jan, Bourbons and Palms Feb, the Bronze were the last to lay