Can forced air heaters trigger allergies?

No. No, but as a simple precaution you should change your air filters at least every 3 months. If your child develops dust mite allergy, then dirty air vents can be a trigger of allergy symptoms, but forced air heaters in themselves will not trigger allergy per se.
Yes. Many people who have allergies are allergic to house dust. Running the forced hot air can circulate this in the house. Using a good filter in the heating system and using an air purifier in the child's room might help reduce some of the exposure.