Simply answer: No. You do not have to claim the dependent care credit.
By why are your asking? There is no reason not to, other than you must tell the IRS who you paid. If the other person is planning on claiming the income, that shouldn't be a problem.
@errios1 said "I know they have to claim it".
Not necessarily, depending on the details. If you don't claim the dependent care credit, the family member can usually treat the money as a gift and not income.
I didn't have childcare that I paid on my 2019 taxes. When I went through to get them done quickly this time around I neglected to put it in (my fault for rushing). Didn't know if it was something I absolutely had to claim and how much of a headache an amendment would be if I did have to. I know a few years ago when I did it was like $100 bucks difference.
Q. Is it something I absolutely had to claim?
A. No.
Q. How much is it worth?
A Simple answer: about 30%* of what you paid (up to $3000, for one kid, $6000 for more than one), but not more than your tax liability (line 16 of form 1040). Unfortunately, taxes aren't simple. So, it could be different.
* See the table on form 2441 https://www.irs.gov/pub/irs-pdf/f2441.pdf