You and your soon-to-be-ex-husband disagreeing about this has nothing to do with the cough, so don't react to that. That's a whole other question!
My take on the symptom business is that if the child has a temperature of 101 or higher, or has a temp lasting more than 36 hours, or is not acting like himself, or looks sick (pale, green, etc.). call the doctor's office and ask if this is worth bringing him in for. You don't mention how old your boy is; an infant needs to be looked after sooner than an older child because you can probably make some educated guesses with an older one, who can communicate with you a little better.
As I recall, if any of my children were going to get sick, it was *always* at 5:05 p.m. on Friday afternoon, just after the doctor's office had closed for the weekend. So I had to check the symptoms and see if we could wait until the next Monday. Sometimes everything was fine by Monday! If not, I would call and ask questions.
What is the severity of the cough? If it's just a little one, and your son doesn't seem to be bothered by it, I might watchfully wait a day or so, but if it goes on much longer I would call. If he's "coughing his guts up," as the inelegant phrase goes, I would think of calling right away. Doctor's staffs usually know if an allergy or anything else is going around.