Your son is your dependent and he should not file his own independent return for the stimulus payments, strictly speaking, assuming your son is under age 24.
If your son is over 23, it is a different story.
1) You are the recipient, the 1099-Q should go on your return if you choose to report it;
2) Your son cannot report the 1099-Q on his return as he is not the recipient.
You may not need to enter the 1099-Q on your return if all the distribution went to tuition and other educational expenses.
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