I think that this is Where We Belong by Emily Giffin, which has the birth mother as the protagonist instead of the girl.
From the summary on Goodreads:
Marian Caldwell is a thirty-six year old television producer, living
her dream in New York City. With a fulfilling career and satisfying
relationship, she has convinced everyone, including herself, that her
life is just as she wants it to be. But one night, Marian answers a
knock on the door . . . only to find Kirby Rose, an eighteen-year-old
girl with a key to a past that Marian thought she had sealed off
forever. From the moment Kirby appears on her doorstep, Marian's
perfectly constructed world—and her very identity—will be shaken to
its core, resurrecting ghosts and memories of a passionate young love
affair that threaten everything that has come to define her.
Her daughter Kirby, is very fond of playing the drums. When she later meets her birth father, who is also a musician, they bond by playing a song together - "Small Town" by Pearl Jam.