There are food allergies and there are sensitivities. I am sensitive to most nuts. That means I can eat SOME, but not much before I have reaction. Generally, I tell people I'm allergic, then they won't serve them to me.
However, if I even come within a foot of a cucumber or melon, let alone eat any of it, I start having a reaction.
"Allergy season" is generally when things are blooming and sending off pollens and such, which are some of the most common allergens. But, that's for someone with pollen allergies. If you have other allergies, "allergy season" is pretty much all year round.