Does my girlfriends son qualify to be my dependent?

able to claim your girlfriend's son  as a Qualifying Relative dependent if:

• You provided more than half of his support.
•  He made less than $4,050 in gross taxable income. (Social Security income generally doesn’t count here.)
•  He lived with you (365 days in the year) or they are related to you.
•  He is not a dependent on someone else’s taxes.
•  He is not preparing his taxes with a spouse (married filing jointly).
•  He is a U.S. citizen, resident alien, national, or a Canadian or Mexican resident?

You can’t claim a dependent if you are a dependent on someone else’s taxes

 See Rules For Claiming Dependents