No. As @Hal_Al explained the taxpayer is responsible for determining the taxable amounts and you cannot double dip. You must determine if there are enough qualified education expenses for the AOC and the ESA. So you should reduce the education expenses available for ESA on the students return by the amount used for the AOC on the parent's return. If there are enough expenses left over including room and board according to your records to cover the ESA then you don't need to report the 1099-Q. Keep your calculations with your tax records should the issue ever come up.
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