No, you don't need the 1099-INT.
If the UTMA is set up with your son's SSN, then any income would be reported on his tax return, if he's required to file one. And, if this $345 was his only income, he wouldn't need to file a return.
Please see the TurboTax FAQ "Do I need to file my own taxes if I'm a dependent?" for information on what your son's filing requirements would be.