In fact, the frost has nothing to do with the Moon. The fact that the Moon is visible at night indicates that the sky is clear and cloudless. Under these conditions, the heat of the ground (accumulated during the day) can rise high in the sky unimpeded. This causes colder nights than if the sky was overcast, and can cause frost.
People in olden times had noticed the correlation between these two phenomena: the Moon and the young shoots, which were seen as though scorched red by the frost. But there is no cause and effect link.
There can be nights when the Moon is visible but there is no frost (if it has not been too cold). And a new Moon will not be visible, but the sky may still be clear and can be frosty.