Heat exposure after FUE Facial Hair Transplant. Would this have any effect?

Unless you burned your face on the BBQ grill, it should be OK. If you are concerned you should follow up with your doctor

In all likelihood there was little effect on the grafts as you most probably had the most heat wave on your face and not the new grafts.  Follow your routine as prescribed by your hair transplant doctor or plastic surgeon.  I doubt there will be any adverse effect from the bar-b-que as long as there was no singed hairs.

The only thing that is to avoided in the recipient area following transplant is the heat only. I don't think there is anything to be concerned as the heat exposure was very momentary.

This type of exposure should not impact the transplanted grafts. You should always be careful with anything you do for a few weeks. 

From your discription, you should be fine.  However,  guard your recipient grafts from extreme heat or sunburn.