No. In order to claim an exemption, the following must apply:
There must be a qualifying relationship, or the child must have lived with her for the entire year.
Your child, stepchild, foster child, or a descendant of any of them (for example, your grandchild). (A legally adopted child is considered your child.)
Your brother, sister, half brother, half sister, stepbrother, or stepsister.
Your father, mother, grandparent, or other direct ancestor, but not foster parent.
Your stepfather or stepmother.
A son or daughter of your brother or sister.
A son or daughter of your half brother or half sister.
A brother or sister of your father or mother.
Your son-in-law, daughter-in-law, father-in-law, mother-in-law, brother-in-law, or sister-in-law.
The girlfriend (if they are not married) does not have a qualifying relationship The child lived with you for 4 months (and not with her for the entire year). Therefore, the only way she could claim the exemption is if he lived with her for the entire year.