Can you damage hair by brushing it after a FUE hair transplant?

It is best to ask your surgeon about post operative care as each doctor has their unique regimen.  Brushing your hair does not damage grafts.

You are welcome to brush your hair - this will not affect your results or damage the planted hairs. 

When you first have FUE hair transplant your hair roots sit freely in your scalp. At this stage they can fall out with minimal effort, so you have to be very careful. Over the next few days these embed into the scalp making them more secure. By day 5 they would have embedded well enough that our clinic allows light washing of the transplanted hairs. By day 14 we allow our patients to resume all normal activities where hair 'popping out' is no longer a concern, such as weight lifting and grooming. This is in the belief that the transplanted hairs have fully embedded. Of course it is important to be cautious, but this is a safe time to start brushing your hair gently and building up slowly as you feel more confident. By 4 months your hair roots have well and truly integrated with the surrounding skin, so there is almost no risk of damage to the transplanted follicles.
Be aware that for any hair, grooming is associated with damage to the cuticle layer of the hair shaft which over time may impact the quality of the hair itself, i.e making it dry.