Cleopatra son by Julius Caesar?

Yes, she did. They fell in love after she snuck into his office through a rolled-up carpet she had delivered to him as a gift. Her intentions were to talk to him about the future of Egypt, but she fell in love instead. They had a son a year later, named Caesarion, or 'Little Caesar'. When Octavian came to conquer Egypt, Cleopatra sent Caesarion away to India to keep safe until he could gain the support to regain his throne. Unfortunately, Caesarion was murdered at sea before he made it to India.