If this person is sitting directly in the line of fire of the aircon, perhaps the temperature and flow of the air is aggravating a pre-existing tendency towards asthma. It could also depend on the proportion of fresh air taken into the system, a larger part of the air in any such system being recycled. Try moving desk and see if that makes any difference, if not then occupational health may need to be involved to assess air quality. You do not say if anyone else there is similarly affected.

Where I worked previously we had issues like this. It seemed to be a combination of both the above issues plus when we brought someone in to test the air, we found that during the cold weather windows and doors were kept firmly shut and actually resulted in a high CO2 environment. One person we just needed to move to another desk in the room (drying out of mouth and throat), but another we had to actually move to another room. Opening windows on a cold day is always good fun. The offices were very old and wooden construction, god knows what was living in the cavities, we spent loads of money trying to find out. Even had all the movable partitions cleaned (you know the sort 4 feet high and covered in material). Eventually a new set of offices with their own unique issues came along.