When talking about how everyone knew Amy was pregnant, they bring up all the typical TV tropes of concealing an actor's pregnancy, when their character is not pregnant. All were used when Melissa Fumero was pregnant during production of season 3 of the show, something viewers were wise to, which this scene pokes fun at.
Jake's grandfather is an admiral in the Navy. Andy Samberg's maternal grandfather was in the Navy and stationed in California during WWII where he met Andy's grandmother.
When Captain Holt is helping Lieutenant Jeffords with playing the flute, the scene mirrors many scenes from the movie Whiplash (2014) in which J.K. Simmons plays a main character. Simmons was also in the previous episode of Brooklyn Nine-Nine as Detective Dillman.
In the pub where Jake meets his dad, on the wall is a poster that says "Santos/McGarry." These are the names of candidates for President and Vice President in the final season of The West Wing (1999). Bradley Whitford, who plays Jake's dad, stars as the campaign manager for Santos, played by Jimmy Smits - who incidentally plays Amy's dad.
Terry Crews (Jeffords) took flute lessons for 8 years as a child after receiving a flute from his great aunt.