For the most part, no. "Lunar twilight" basically starts when the moon is just under the horizon / ends right after the moon sets. See the question "Lunar twilight and sixth magnitude stars" on Physics.SE:

However, even when below the horizon, the Moon shines brightly enough to interfere with stargazing. What are the equivalent twilight angles for the Moon? I realize that even the full Moon overhead isn't bright enough for civil twilight, so my real interest is in astronomical twilight.

Lunar twilight is similar to solar twilight, but it is much more bleak. The illumination from the moon varies from 10−9 (new moon) to 2⋅10−8 (full moon) from the illumination from the sun at the same point in the sky. Therefore the illumination from the full moon high in the sky corresponds approximately to the middle of nautical twilight. And lunar twilight ends virtually when the moon gets under the horizon.

If your shutter is exposure is more than a few minutes (such as when using a tracking mount) and you are shooting very close to moonrise, then I would expect some horizon "bloom", much like sunrise, although substantially less pronounced. Similarly, if you are taking star trail images that end shortly before moonrise, you might have some horizon bloom.

But if aren't tracking, and are looking for trail-less star images, then your shutter won't be open long enough to have any appreciable atmospheric bloom.