Yes, when a queen bee mates with drones, she becomes fertilized and can lay eggs. These eggs develop into the different types of bees in the colony, including workers and drones. Worker bees are indeed the daughters of the queen, making them her offspring. The queen can lay thousands of eggs in a single day, and those eggs develop into the various roles necessary for the colony's survival, including foraging for nectar and pollen, building and cleaning the hive, and caring for the queen and her young.
However, drones, which are male bees, are also offspring of the queen, but they are primarily produced for mating purposes. So, in a honeybee colony, most of the bees that work (the worker bees) are the queen's daughters, while drones are typically her sons.