At 3 months old, AJ starts to smile.
Or
At 3 months old AJ starts to smile.
At 3 months old, AJ starts to smile.
Or
At 3 months old AJ starts to smile.
According to the Purdue Online Writing Lab (Purdue OWL), you should not use a comma after a brief introductory prepositional phrase.
On the page Commas after Introductions, under the section labeled "When not to use a comma," they state:
Is it a single phrase of fewer than five words?
Since "At three months old" is a single prepositional phrase of only four words, you do not need a comma according to this style guide.
Note, however, that this is more a matter of preference than a strict grammatical rule. Other style guides may vary.
It depends on what will follow he sentence. Without a comma it would sound good to my ear to hear "At 3 months old AJ starts to smile. At 4 months he started to laugh" It would be part of a progression. If the next sentence expanded on things like when AJ would smile then to me it would sound better with a comma. For example, "He would see his mom and smile."