Same first name...

My husband and I have been going through potential baby names, and picked a first and middle name combination for a girl that we absolutely love. There is a good chunk of extended family that my family has absolutely nothing to do with; after we fell in love with the name I remembered that I have a cousin whose daughter has the same first name, and they call her by a shortened nickname version.

Does technically having a living blood relative with a name make it unavailable?