Michael J. Fox Set Up Moon Unit Zappa and Woody Harrelson in the '80s. It Ended in the Shower (Exclusive)

Moon Unit Zappa, daughter of rock legend Frank Zappa, grew up surrounded by celebrities. She was still in her early teens when she appeared on Late Night with David Letterman with her dad Frank to promote their 1982 collaboration "Valley Girl," a novelty song featuring Moon on vocals that became her father's only pop hit.

Moon, 56, recalls various brushes with current and future stars in her new book Earth to Moon, A Memoir. Erik Estrada, Emilio Estevez, Whitney Houston and Howard Stern all make cameos. But she saves the best celebrity encounter for Chapter 40, "One Fresh Raisin," which kicks off with her good friend and Family Ties star Justine Bateman calling Moon on the phone.

"Mike wants to set you up with a friend of his," Moon recalls Bateman saying to her in the book, referring to her Family Ties costar Michael J. Fox. "He's suggesting we meet at a billiard place."

The not-so-blind date ended up being with Woody Harrelson, who was at the time appearing on the hit NBC sitcom Cheers. It's not exactly love at first sight, but a game of pool and a dinner invitation follow. They ended up going on four dates — including one where he used her index finger to pull a piece of food from between his teeth — before things ended awkwardly in the shower after he asked her for a raisin.

"That story is so funny, because in some ways, it's about Woody, but in some ways, it's really more about me and the fact that I'm raised in such a way that those red flags, I just keep ignoring, and ignoring and ignoring until I look like the weirdo," Moon tells PEOPLE.

"And so I think that that's why that story is funny to me," she continues. "Maybe to other people, it'll be a story of a celebrity dating situation, but for me, it's to show how much latitude I gave to people as a result of my upbringing."

Moon's upbringing is a major touchstone in Earth to Moon, A Memoir. She had a difficult relationship with her father, Frank, who died in 1993 of prostate cancer. He was often not at home because he was on the road touring. And when he was home, he was emotionally distant.

Her relationship with her mother, Gail Zappa, was even more complicated. With Frank often gone, Gail, who died in 2015, was effectively a single parent to Moon and her three younger siblings. Moon writes that her mother was emotionally abusive, and, in conversation with PEOPLE, refers to Gail as her "first bully."

Looking back at her ill-fated last date with Harrelson, she sees that her knee-jerk response to him calling out to her from the shower and asking for a raisin — or so she thought at the time — was a symptom of how she was raised.

"Growing up in my family, we had to do exactly what was needed, at exactly that moment, and rush to do it without questions being asked," she explains. "And maybe it was because my mother came from a military background, but we just had this military response, or at least I did, to just get the thing done. And part of that was to, if I could do things well and do them quickly, then there was a chance I wouldn't get harassed."

"And so part of this was a trauma response to just getting things done as fast as I could, at the highest level, so that I could at least stave off some attacks," she continues. "It carried over into my personal life, where regardless of the quirkiness of the thing that somebody was asking for, I just would go against every instinct and just deliver."

As for Harrelson, things may not have worked out between them, but some 35 years later, there's no bitterness or resentment. "I've run into him a few times," she says, "and we've had a few laughs."

Earth to Moon, A Memoir is published by Dey Street Books, an imprint of HarperCollins, and is available now, wherever books are sold.