There is actually no specific time or day when Allah will accept your Dua
because Allah is always with us in our hearts.If we are good,honest,faithful and offer prayer 5 times a day and can keep Allah happy with our good deeds,then surely He will help us and give us what's best for us He thinks.And sometimes He might also give us what we want so it depends upon Him what He chooses best for us.You can ask for anything to Allah anytime of the day.But I have heard that doing Dua after each prayer session is the best.
__________________________________________________ ____________
Search Sahih Muslim
Quran English Translation