Yes, cats can have allergies, but I've never actually heard of allergies making a cat cough (I don't even know if that's a possibility). Typically, allergies cause skin and ear problems. Asthma, hairballs, and airway obstructions can certainly cause coughing, though.
my previous cat used to cough after he drank. i took him to the vet when i first got him and we ruled out everything we could. the vet said it was probably just some kind of malformation that he was born with. He would hide to cough too, like he didnt want me to see him doing it. If i found him and he saw me, he would stop. It never caused any issue, and he did it all his life, but I would definately take your cat for a checkup just to rule out any medical problems.
A little while back Meeka was coughing about once a day. I thought she was trying to bring up a hairball but after about 4 days, I got really worried nothing had come up and took her to the vet. She had an infection. She was given a shot, and some antibiotics and it had cleared up within about 5 days after that. Samson started coughing the day after we took Meeka to the vet so the next day, in he went as well. It was obviously contagious and he was given the antibiotics (no shot for him) and within a few days, he was better as well.
They were both otherwise fine within them selves, eating, drinking, playing with the kitten. Nothing changed except they had abit of a cough. It was an infection though so I highly recommend you have it checked, even if she seems fine otherwise.