Am I asked to pick a specialization when a doctor levels up?

I'd like to choose differential diagnostic perk, but he's already level 3 and no spec yet to choose and I don't know if I'm asked when he levels up maybe to 4 or 5?

Second Specialization Perk is unlocked on Level 4 (Fellow), when you didn't change the notification you will get a pop up where you can select the second perk. You Can always change perks, but beware if you do so, they will start at 0 in that perk and it will override the progress on the other perk.