1 month old baby. her right eye is just producing tears every now and then but no redness. it looks tired too. is there an issue?

Check with peds: Probably she has blocked tear duct. Using clear water on a clean, soft cloth, gently wipe any discharge out of your baby's eyes. Do this several times a day or as often as necessary. The tear ducts lie just beneath the nasal corner of the eyes. Gently massage this area moving your fingertip in a semi-circle from the corner of the eye inward toward the nose. But for sure bring her to see peds.