Each of my children have a UTMA account. They also both received a 1099-DIV for approx $13 each for 2017. Do I need to file a return for each of them?

No.  As long as it is just dividend income they can have up to $350 without filing a return (and you don't have to include it on your return, either).

Actually, the can have up to $1050; but that includes income (interest, dividends and capital gains) within  the UTMA account.
UTMA accounts are not like IRA or 529 plans. The tax is not deferred. It's only shifted to the child's lower tax brackets.

You do not report his/her income on your return*. If it has to be reported at all, it goes on his own return. If your dependent child is under age 19 (or under 24 if a full time student), he or she must file a tax return for 2017 if he had any of the following:

1.         Total income (wages, salaries, taxable scholarship etc.) of more than $6,350 (2017).

2.         Unearned income (interest, dividends, capital gains) of more than $1050 (2017).

3.         Unearned income over $350 and gross income of more than $1050

4.         Household employee income (e.g. baby sitting, lawn mowing) over $2000 ($6350 if under age 18)

5.         Other self employment income over $400, including box 7 of a 1099-MISC

 

Even if he had less, he is allowed to file if he needs to get back income tax withholding. He cannot get back social security or Medicare tax withholding.

 He doesn’t get his own $4050 exemption (deduction), when he files. In TurboTax, he indicates that somebody else can claim him as a dependent, at the personal information section.

*If his only income is from interest and dividends, Alaska PFD or capital gains distributions shown on a 1099-DIV, there is a provision for entering it on your return, using form 8814. Enter at Less common income / Child's income.

But, it is usually best for him to file a separate return, as qualified dividends and capital gain distributions could be taxed at a higher rate on the parent's return.