To answer your specific question about Form 1099-Q it isn't taxable, so you don't need to enter Form 1099-Q.
Yes, in your situation, if your daughter is filing as if she is not a dependent, your daughter can also claim the student loan interest that was paid.
If you pay her college and room and board, I think you should claim her as your dependent. Does she work? How can she support herself if you pay for everything? Perhaps you should talk to someone about whether you should be claiming her, but you don't have to worry about the 1099-Q.
Usually the parent claims the college student and also uses Form 1098-T (even when it is issue to the student) to get an education credit which can lower the parents tax liability and also inflate the parents refund.
If the student does not have a high income, they probably don't have a tax liability and the credit is reduced.
You might ask your daughter to talk to the financial aid office at the school and they might be able to explain what's best for both her and you.
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