Your first sentence is not quite right. I think "She quietly fell into a dream on the beautiful night." would sound better, or you can still say "Quietly she fell into a dream on a beautiful night. Your second sentence is perfect. Other examples along these lines would be:
The cat walked quietly up the street.
I carefully stacked the plates.
A puppy was playfully chasing it's mother.
Hungrily, the little girl took the cookie.
I used to speak Spanish so easily.