Absolutely you can get allergies from air I wake in the mornings with sore /itchy eyes, sneeezing, sniffing .
whilst it could be a summer cold, it IS possible your air conditioning may have caused or contributed to your symptoms. Here are a few suggestions to minimise the risk of symptoms from your air conditioning;
it may be dust or pollen, stired up by the air conditioning. if you check/clean any filters the air conditioning system may have, it could help.
if you increase the frequency of dusting, there will be less dust to be circulated in the air make you ill.
there is a chance that even if you have not had hayfever previously, it is possible to develop it, especially now what-ever pollen is in the air, is being well circulated.
also an air purifier should help remove pollen/ dust/ WHATEVER, that is causing it, and should be filtered out.
if these don't help your symptoms enough, then it may be worth visiting a doctor or chemist for some relief.