It is possible if the hay is dusty or moldy it is causing irritation and making him wheeze. Having the vet out to confirm and medicate if necessary as well as rinsing his hay as you feed can help if he is being fed by the flake.

 

If he's wheezing, you need him examined by a vet. If he has asthma, switching onto hay would absolutely cause a flare up. By the time you can notice signs of wheezing or laboured breathing, there is already constriction and damage happening in the lungs. Your should be able to tell you if this is an allergic reaction, asthma, a pneumonia, ect.

How are you feeding the hay? Round bale, netted, squares, ect? If it's possible to soak or steam the hay, do that.

My guy got a shiny new asthma diagnosis for Christmas. He's been taken off the round bale eats compressed square bales now. It's not practical to soak hay for most of the year, so I chose the more expensive compressed hay because it's very low dust. He had a BAL done and a trachea scope to confirm the diagnosis and rule out pneumonia, then did an aggressive treatment with two inhalers 2x a day for 2 weeks, then 1x a day for 1 week.

 

You need a Vet. Healthy horses do not wheeze. It could be dust mold from the hay or it could be a pneumonia or copd/heaves.