If your nascent transcipts/amplicons all contain deoxyuracil instead of deoxythymidine, UNG should work, but if your mastermix hasn't replaced deoxy-T's with deoxy-U's (in the dNTP's used in the mastermix) UNG will be of little use as it only works on (degrades) sequences containing deoxy-U's.

The worry here is whether you or a co-worker has contaminated the room with same-target amplicons (perhaps by taking the cap-strips off of an already-run qPCR plate) that were amplified with a dU-containg mastermix (replacing dT's amongst the dNTPs used). You will see that some mastermixes do not contain dUTPs. In which case, the damage has already been done (if the room is swirling with amplicons containing dTTPs for some unknown reason).