Maybe. Young babies can have allergic nasal inflamation (rhinitis) with constant runny nose. A 6 month old can also have multiple cold during the winter and have an almost constant runny nose from fall to spring. Speak with your pediatrician about the appropriate dose of an antihistamine to use for your baby. If control is inadequate consult an allergist.
Possibly. Allergies can occur at any time and are generated from a specific antibody against protein. Though it is possible for a six month old child to have allergies, it is not very likely. Allergic antibody from the mother is present at birth and usually lasts until 3 months of age, sometimes longer. Infant ige (allergic antibody) is not usually produced in suffiencient quantities until after 6 months.