Hi there --
While this definitely could be related to your hay fever and history of nosebleeds (probably also related to each other), you may even have asthma, which often presents insidiously and without loud, obvious wheezing.
You definitely should have at least a general medical exam with an eye toward uncovering any possible lung problem, and in particular to rule out asthma, especially exercise-induced type.
I hope this is helpful. Good luck to you. Please follow up with us here as needed.
John Kenyon, EMT, CCT
Non-invasive cardiology tech, Emergency and Critical Care technician, Critical Incident Stress Mgmt. specialist