Thanks for using JustAnswer.com, I'm Dr. Dan and I'll try to help you.
Yes a cockatiel can produce eggs without mating or without a male bird in the house. The conditions need to be right to trigger the bird hormones and include, interaction with you, humidity, temperature, daylight, level of fat in the food, etc.etc. You can feel free to remove the egg as it really has no affect on the laying of future eggs. If your cockatiel continues to lay several eggs then you should set up an appointment to make sure your bird can physically handle all the egg laying that is going on. Some birds can lay enough eggs to make them sick. If your bird lays more than 3 eggs then it's time to see the vet. I hope this helps best wishes.