A 529 plan generally has two names associated with it. First, there's the plan administrator. IN your case, I would expect the plan administrator to be grandma, since she's the one that opened the account.
Second, the plan will have a named beneficiary of that plan. The beneficiary of the plan is usually the student. But it could be the parent of that student. The named beneficiary is the one who will receive the 1099-Q for any funds withdrawn from that account, and the SSN of that named beneficiary will be on that 1099-Q.
Regardless who whose SSN is on the 1099-Q, if you qualify to claim the student as a dependent on your tax return, then you are the one who will report the 1099-Q regardless of whose SSN is on that form. (your SSN, or the SSN of the dependent student you're claiming).
There are exceptions where the student would report it, as clarified in my initial response above.
