No, the cost of private school (K-12) doesn't qualify for any education deductions or credits on your federal return. A few states allow deductions for K-12 education and/or home schooling expenses. If your state is one of them, we'll ask you about it when we get to your state return.
If your child attends a daycare program before or after school, whether at a public or private school, you may qualify for a child care credit. Some private school tuition for special needs students may be eligible for a medical deduction.