Speaking as a Quaker (with Quaker roots going back to the late 1600's), we still sometimes use the term 'birthright Quaker' for someone born of Quaker parents, though it's not official. We have some discomfort with the term, since we don't want to convey that any special virtue (or vice) gets passed in a family, and we definitely don't want anyone to think we believe someone is better merely because of their family background. (We tend not to believe in 'original sin' and don't sprinkle babies or adults. It's a big difference from the Puritans, who not only believed in original sin, but also tried to beat it out of their children, criminals, and wives.
Although Quaker parents traditionally stand up in meeting with the new baby (as mine did) and publicly announce their intentions to raise the child in the Society of Friends, this is merely a custom and conveys no mystical significance or 'hocus-pocus.' There is no defining Quaker moment until the request for membership. "Birthright' may or may not imply that a person never took the next step to request membership as an adult. (I did this around 16.) The other writers are quite correct in mentioning that things vary over geography and time.
I think you are perfectly justified to speak of someone born to Quaker parents as a birthright Quaker, understanding that this is essentially a folkway, but conveys no official sanction.