No, you have no reporting requirement as the custodian. The income from UTMA accounts is the named child's income and is reported under his/her Social Security number.
For the child's reporting requirements, please see the following, which is written from the parent's perspective.
Your dependent child's income from investments is taxable
income and must be reported if it exceeds the filing threshold. If your
child's only income is unearned income not exceeding $1,050, it doesn't need to be
reported.
The Kiddie
Tax is applied to the amount of your
child's unearned income that exceeds $2,100. The Kiddie Tax rate is your
marginal tax rate (the highest rate applied to the last dollar you earned)
which could be as high as 39.6%.
There are actually two ways to report and pay this tax: either
by filing a tax return for the child alone or by including the income on the
parents' tax return. Either filing method will yield the same tax.
Please refer to this TurboTax FAQ for a thorough
discussion of How
do I report and pay the Kiddie Tax?