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She may associate a cough with a negative experience that had nothing to do with the coughing but is just connected in her mind.
You could try a desensitization program with her where you fake a throat clear or a cough as quietly as possible and then give her a small treat that she adores.
Sorry dog emergency here will be back in a minute to finish this!
Sorry about that had to sort something out with the critters here - I swear they wait until I'm in the middle of an answer like kids do when you are on the phone!
The purpose of the desensitization is to associate the sound with a positive thing for the dog. So if the dog has a treat it love such as a bit of freeze dried (or cooked) liver, chicken, cheese or a doggie treat you give a tiny bit of that after making the sound the dog is afraid of to change the dog from 'scary sound run for your life' to 'hmm there's that noise again where is my treat?'
A cough or throat clear can sound a lot like a growl to a dog and that might be a part of the problem.
If the dog acts afraid don't comfort her, instead laugh and tell her she is being silly and act like its nothing. How you react to her fear reaction can change her fearfulness or make it worse. While we humans respond to comforting sometimes it sounds to a dog like we are also distressed and instead of hearing 'its OK' they hear 'oh oh oh its terrible and you are right its something to fear'. Has to do with the tone of our voice and the natural sounds dogs make when distressed as pups. So you want to avoid that and instead project that the sound isn't anything important at all.
Some dogs are sound sensitive and may need help in overcoming a fear of a sound. If home training doesn't work you may want to talk to your vet about using melatonin which can help with sound phobias in dogs or using a medication to take the 'edge' off the dog's fear. In both cases you continue training but with the added aid of the reduction in fear due to the product being used.
I did check the extra info and thanks! It could be she associates coughing with high stress as I'm sure you were all upset over the other dog's illness.
Hope this helps you!