The fee will be the usual fee to cancel the infant economy seat, and upgrade the adult economy seat to business, and add a lap infant (usually 10% of the business fare plus any taxes and airport charges). There is no special deal.
If Qatar is having a sale now it would be cheaper to just buy the adult business seat plus lap infant now (or do the upgrade) rather than seeing what the price would be at the airport. It will most likely be higher.
As for whether to do it, I would say yes if you have the money; it's certainly more comfortable for the adult. You could get a bassinet seat and the baby could sit or sleep there for a few hours, but it depends on the size and weight of your baby. My son is 6 months and he is too big and heavy for most airline bassinets now, so we just have to hold him.
I agree. The agent at check-in won't be empowered to do anything other than reverse the existing bookings (which may or may not have penalties) and upgrading these (if the fare class allows it). I have my doubts as to whether the final price for this will be something that falls into the 'I was expecting this' category.