It's the inability to digest the protein in milk. Are asking about "lactose intolerance" in your milk or the cows milk you drink ending up in your milk?
I experienced this with my last baby: I am not able to eat any milk products because of her intolerance.
If the baby is having a reaction to the milk that you are drinking, he could end up with colic, gas pains, green stools, explosive gas when he makes a bowel movement, and bloody stool in advanced cases. It will take 1-2 weeks for the symptoms to go away depending on how long you have been drinking the milk and how intolerant he is to the milk. He could also end up with eczema from the milk.
An allergy to your actual milk is very very rare, but it does happen. But normally you can fix whatever the problem is with a gradual elimination of possible problem foods from your diet.