I'm the account owner of a 529 for a grandkid. I took a withdrawal and paid the school. Can I prove it was a qualified education expense and not get a tax and penalty?

Q. Can I prove it was a qualified education expense and not get a tax and penalty?

A. You don't have to "prove it". If you know it's so. Just don't enter the 1099-Q

The 1099-Q is  only an informational document. The numbers on it are not required to be entered onto your (or your student's) tax return.  You can just not report the 1099-Q, at all, if your student-beneficiary has sufficient educational expenses, including room & board (even if he lives at home) to cover the distribution. When the box 1 amount on form 1099-Q is fully covered by expenses, TurboTax will enter nothing about the 1099-Q on the actual tax forms.  You would still have to do the math to see if there were enough expenses left over for the parents to claim the tuition credit. You also cannot count expenses that were paid by tax free scholarships. 

References:

1. On form 1099-Q, instructions to the recipient reads: "Nontaxable distributions from CESAs and QTPs are not required to be reported on your income tax return. You must determine the taxability of any distribution." 
2. IRS Pub 970 states: “Generally, distributions are tax free if they aren't more than the beneficiary's AQEE for the year. Don't report tax-free distributions (including qualifying rollovers) on your tax return”.
3. ("IRS Publication 970, Tax Benefits for Education states: If the entire 1099-Q went to qualified expenses, room and board, tuition, etc then you do not need to enter the form." 

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Qualified Tuition Plans  (QTP 529 Plans) Distributions

General Discussion

It’s complicated.

For 529 plans, there is an “owner” (usually the parent, but can be a grandparent), and a “beneficiary” (usually the student dependent). The "recipient" of the distribution can be either the owner or the beneficiary depending on who the money was sent to. When the money goes directly from the Qualified Tuition Plan (QTP) to the school, the student is the "recipient". The distribution will be reported on IRS form 1099-Q. 
The 1099-Q gets reported on the recipient's return.** The recipient's name & SS# will be on the 1099-Q.
Even though the 1099-Q is going on the recipient's (your) return, the 1098-T should go on the parent's return, so they can claim the education credit. They can do this because he is their dependent.

They can and should claim the tuition credit before you claim the 529 plan earnings exclusion. The American Opportunity Credit (AOC or AOTC) is 100% of the first $2000 of tuition and 25% of the next $2000 ($2500 maximum credit). The educational expenses you claim for the 1099-Q should be reduced by the amount of educational expenses they claim for the credit.
But be aware, you can not double dip. You cannot count the same tuition money, for the tuition credit,  that gets you an exclusion from the taxability of the earnings (interest) on the 529 plan. Since the credit is more generous; use as much of the tuition as is needed for the credit and the rest for the interest exclusion. Another special rule allows them to claim the tuition credit regardless of whose money was used to pay the tuition.
In addition, there is another rule that says the 10% penalty is waived if you were unable to cover the 529 plan withdrawal with educational expenses either because he got scholarships or the expenses were used (by the parents) to claim the credits. 

Total qualified expenses (including room & board) less amounts paid by scholarship less amounts used to claim the Tuition credit equals the amount you can use to claim the earnings exclusion on the 1099-Q. 
Example:
  $10,000 in educational expenses(including room & board which is only qualified for the 1099-Q)

   -$3000 paid by tax free scholarship***

   -$4000 used to claim the American Opportunity credit

 =$3000 Can be used against the 1099-Q (on the recipient’s return)

Box 1 of the 1099-Q is $5000

Box 2 is $2800

3000/5000=60% of the earnings are tax free; 40% are taxable

40% x 2800= $1120

There is  $1120 of taxable income (on the recipient’s return)

**Alternatively; you can just not report the 1099-Q, at all, if your student-beneficiary has sufficient educational expenses, including room & board (even if he lives at home) to cover the distribution. You would still have to do the math to see if there were enough expenses left over for you to claim the tuition credit. Again, you cannot double dip!  When the box 1 amount on form 1099-Q is fully covered by expenses, TurboTax will enter nothing about the 1099-Q on the actual tax forms. But, it will prepare a 1099-Q worksheet for your records, in case of an IRS inquiry.

On form 1099-Q, instructions to the recipient reads: "Nontaxable distributions from CESAs and QTPs are not required to be reported on your income tax return. You must determine the taxability of any distribution." 

***Another alternative is have the student report some of his scholarship as taxable income, to free up some expenses for the 1099-Q and/or tuition credit. Most people come out better having the scholarship taxable before the 529 earnings. A student, with no other income, can have up to $14,600 of taxable scholarship (in 2024) and still pay no income tax.