Does the concept of Brahma muhurta (time of Brahma) exist in Buddhism?
I am unable to find a reference, but I have heard at a Dhamma sermon that the Buddhist monks are advised to get up at 4 in the morning, then clean the mouth and have a light soup which consists of salt, water and may be some other ingredients.(A light soup with salt in it) After that monks are advised to learn the Dhamma, may be memorizing, understanding etc. So yes it is advised to get up at 4 am and study but I am not sure whether the term Brahma Muhurtha is used in that scenario. Outside the Dhamma, many studies have been done on morning people and they have been found to be more successful, proactive and happier. 

http://www.inc.com/news/articles/2010/07/research-says-morning-people-are-more-proactive.html
http://www.prevention.com/health/sleep-energy/study-finds-morning-people-are-happier

Laymen also can follow these steps and can become happier and more successful in reaching their visions.