I am really not sure what I would do. If it isn't red or full of pus I wouldn't think infection. Allergies seem a bit odd with just one eye. Sorry, I'm not sure. I have a good goat vet, I would probably ask her when I should worry.
Can you pull the eyelid down to check for a foxtail or a particle of a plant/debris? The little experience I have with watery eyes, is first, foxtails. Had 2 different goats with a foxtail in the eye this year. Second, super dusty air, from shaking the walnut trees/hay/ and just yucky mucky air. Eyes got better immediately once the foxtails were removed, one with antibiotic ointment to assist in the healing for what appeared to be the start of an ulcer. The others clear up on their own in a few days.
Can you clean the area and try to flush the eye with plain water? Here is a tiny blurb I found online about goats eyes.
Goats have especially unique eyes. Not only do they lack tear ducts, they also have horizontal rectangular pupils. Some people may find a goat’s eyes startling, because his horizontal pupils stand out against his brightly colored iris. Others find a goat’s eyes fascinating because they allow him to see almost all the way around himself, as well to have very detailed, 3D periphery vision. Unfortunately, like rabbits, goats are prone to some common eye problems. Their lack of tear ducts leaves them unprotected against debris in the air, such as seeds or hay. They’re also prone to less serious issues, such as inverted eyelids. Contact your veterinarian if you notice any excessive watering, bruising, discharge or eye damage.
We live on a gravel road and a couple of the pastures are right alongside it. I try not to use them during the dry dusty times but inevitably dust gets in the eyes sometimes. I see some watery eyes during high pollen and dry dusty conditions. I watch them closely and if they don't clear up by moving them over a couple of days, I'd have to call the vet just to be on the safe side.