Those are the incorrect locks, those are designed for a 350Z. Get the correct ones from the dealer or from Infiniti, and make them give them to you for free. It broke off because the studs on the 370Z are longer than the 350Z, hence the lug bottoms out and the top of the stud puts pressure against the top eventually breaking it off.
Here is a thread on the issue.
Broken wheel stud and lug nut.
Valid part numbers:
999W2-Z3409
999W2-JT000
The correct part will have a flat washer base like the regular lugs.