Yes, it is perfectly fine to color the hair during pregnancy. There is very little of the hair dye absorbed into the system. This small amount of chemical has not been found to be harmful to the growing fetus.
If a woman is still concerned, she could wait until the pregnancy is beyond the third month into the fourth month or the last half of the pregnancy when all of the babies’ organs are already formed.
The fumes of the hair preparation may be a problem in pregnancy, but this should be minimal in a well ventilated area.
The hairdresser may be concerned about how well the hair will take the color because of the changes of pregnancy.
It is important that a pregnant woman feel comfortable in her appearance and if she wants to touch of the color or change her hair color, she should go ahead.