Not in newborn. Lactase which digests lactose or milk sugar essentially always present in newborns and levels only decrease in toddlers at age children would usually be weaned. It is lack of lactase that causes lactose intolerance. About 2-3% of infants are milk allergic so this is a more likely cause of reactions to cow milk. Switch to soy based formula and consult an allergist.
Diarrhea/ rash. If present from birth, this often produces a chronic diarrhea with acidic stools that leave baby with a scorch looking diaper rash focused about the anus. After a while yeast will often join & aggravate the rash.If the diarrhea clears after a week on lactose free formula, with a return after a return to lactose, your most likely lactose intolerant.