Is it possible to know the date a woman conceived, when they go in for the start of prenatal care?

Only an estimate: Doctors can estimate the date of conception, but it is not possible to know the exact date of natural conception.

We can estimate: Conception is normally about two weeks after the start of a period. If an ultrasound is performed early in pregnancy the date of conception may be estimated as two weeks. For example if the ultrasound says that the pregnancy is 8 weeks along, conception could be estimated 6 weeks earlier.