Can birth marks appear three months after birth, especially under the feet?

They can: The term "birthmark" is a generic term for any marking or spot that pops up near birth. It could be a mole, a hemangioma, a cafe au lait spot, or any other variety of marking. Unless the mark is changing rapidly, it should be stable until you can show it to your pediatrician. If it is starting to look suspicious, have it looked at now.