I think you're talking about iMessage?
If you're connected to wifi or data and doing an iMessage (appears in blue instead of green) then no you wouldn't be charged towards texting limit.
I'm not that familiar with how iPhones work but I do think there might be some issues that could arise that would count towards texting limit..
Example if your son is in a group iMessage chat and someone does not have an iPhone that groups chat I think gets sent or received both via iMessage for apple users and sms messages for non apple users... so that might count towards text Usage.
I also believe you can send an iMessage to non apple users and that is sent via SMS as a text (showing the message in green for sms rather then blue for iMessage) that would count towards usage as well.
Hopefully someone that uses iPhones on a daily basis will respond to your question to give you a clear concise answer.
Seems to be good! Thanks for the help and the information.
Can anyone tell me if apple to apple texts count towards his outgoing texting limit?