I received a 1099-DIV for my under-18 child. I am listed as the custodial parent. Do I need to report the Qualified and Ordinary Dividends as reported on the form on my taxes?

You do not have to report the child's investment income on your return. The child can file a return and pay the taxes due, but part of the income may be taxed at the parent's rate (known as the kiddie tax). However, you can elect to report the child's income on your return if you meet the requirements.

You may want to review this:

Tax Topics - Topic 553 Tax on a Child's Investment Income

If the dividend income is more than $950 (with no other income), child is required to report this income.
So if it is less than $950, you need not do anything.
Your U.S. Tax Return