Any time the writers can get Avery Bullock to say something Picard-esque, they will. Best highlighted by his references to "Number One" in One Little Word.
There are also several low-key visual gags which serve as allusions, such as Picard's distinctive fish tank showing up in Bullock's home.
On the flip side, they also rely upon this when Bullock is doing or saying something completely ridiculous simply because the mental image of Captain Picard doing or saying something ridiculous is part of the hilarity.
In one episode, Avery Bullock shows off a device exactly like a Star Trek holodeck. When Stan makes the obvious comparison, Avery angrily remarks that he doesn't even know what Star Trek is.
In one episode, Klaus tries to get Stan to put him back in a human body and says he'll get a job making animal sounds. "Making animal sounds" is pretty much Dee Bradley Baker's entire job description (though Baker does human voices as well; he's even announced game shows).
There was an episode in which Roger disguised as Stan, but found that he's unable to mimic Stan's voice. Another episode had Stan do an equally-terrible impression of Roger. Both characters are voiced by Seth MacFarlane.
In one episode, Steve sings Scott Grimes' hit song, Sunset Blvd. He also says that two movies Scott Grimes was in, Critters and Critters 2, are crappy.
Snot's name could be considered this: his voice actor played Dudley "Booger" Dawson. His appearance also resembles a younger version of Booger.
In one episode, Hector Elizondo appears as a concierge, mimicking his famous role in Pretty Woman. It's even lampshaded to all hell and back.
In "Boring Identity", the doctor that tells Francince of Stan's amnesia looks and sounds like Archie Morris, a character that Scott Grimes played in ER. (Obviously, Grimes voices him here as well.)
Channel Hop: Because of low ratings and FOX's plans to revamp their Sunday night line-up with live-action shows (notably Brooklyn Nine-Nine and Mulaneynote though Mulaney proved to be a failure with critics and in the ratings and has been canceled) alongside the Cash Cow Franchise animated shows they still have (The Simpsons, Family Guy), and Bob's Burgers (which actually has a loyal viewership and critical acclaim, even if the ratings aren't spectacular)note and a new series coming in spring 2015 called Bordertown, made by Family Guy's Mark Hentamenn with Seth MacFarlane as producer, American Dad has moved to TBS in late 2014. Initially, it was going to end its run there by airing all the episodes that FOX never did, but has recently been renewed for another season.
The Danza: Hayley's on-and-off boyfriend (now husband), Jeff Fischer, voiced by Jeff Fischer. His voice actor even looks like a real-life version of Jeff, making him an Ink-Suit Actor.
It's a testament to Seth MacFarlane's vocal talents that his voice doesn't leap out at you as much in this series (or The Cleveland Show, as MacFarlane only voiced Tim the Bear and Dr. Fistnote who were later replaced with Jess Harnell and Bryan Cranston, respectively). Most viewers can figure out that Seth voices Stan note (since the voice of Stan has been used on Family Guy in a couple of parodies of 1950s educational short films, like the one on "I Am Peter, Hear Me Roar" about how men should treat women in the workplace [which includes a lot of emotional manipulation and sexual harassment, as this is the 1950s]), but a surprisingly large number of people aren't aware he voices Roger the aliennote though, if you watch both this show and Family Guy long enough, you can sort of tell that Seth voices Roger because Roger's crying sounds exactly like Peter Griffin's. On top of that, Seth actually broke out his Roger voice during his monologue on Saturday Night Live and a DVD-exclusive special on how American Dad was created and the table read for the second episode "Threat Levels" and Greg the newscaster.
Hayley is voiced by Rachael MacFarlane, Seth's younger sister, which leads to a lot of incestuous undertones as both Seth and Rachel have played characters who get romantically involved (Stewie Griffin and Olivia; a couple episodes of American Dad have had references to Hayley being romantically involved with her father — in "Francine's Flashback," Stan's plan of revenge after Hayley's boyfriend runs off with Francine is for Stan and Hayley to date each other [until Stan realizes that he didn't think the plan through], and "Pulling Double Booty" was all about Hayley dating a man who actually looks like Stan).
Missing Episode: The Christmas episode "Minstrel Krampus" that was scheduled to air on December 16th, 2012 was pulled following the Sandy Hook Elementary School shooting. It would be almost a year later before this episode would air around Christmas time.
The Other Darrin: Reggie the Koala's voice actor changed between the season four episode "Family Affair" and "Wife Insurance" and again in season five. The voices don't sound alike at all.
Duper was voiced by Phil Lewis in most of his appearances. Dee Bradley Baker voices him in "The Kidney Stays in the Picture."
Akiko was voiced by Grey Delisle until "Best Little Horror House in Langley Falls", where she was replaced by Grace Park.
Half the things Avery Bullock says are hilarious simply because they sound so bizarre coming out of Patrick Stewart's mouth.note Picture Picard saying this to his crew and try to keep a straight face. "Gentleman do what ever it takes to empty our coffers: lapdances, champagne dances, shower dances. Oh, the things you can buy for a hand full of bills. It makes me excited. It gives me these chills. They'll be filcher rubs, breeders, hambones and tweeners, zobos and debos and blorps that go 'eenahs'. For a one dollar bill you can pull down their zippers. I am the Snorlax, I speak for the strippers!"
Originally, Laura Prepon (Donna from That '70s Show) was supposed to voice Hayley Smith, but was replaced by Seth MacFarlane's sister, Rachel. Had Prepon stayed, the show would have gotten a lot more unfavorable comparisons to Family Guy as the latter show also has someone from That '70s Show as a cast member (Mila Kunis replaced an uncredited Lacey Chabert on Family Guy and played Jackie on That '70s Show).
According to the commentaries, the original plan for Roger would've been that season 1 would've had him be stuck in the house all the time, but get fed up with it in season 2 and try the disguises. They would've worked until season 3, where Roger gets revealed in public and it becomes a giant media scandal. But by season 4, everyone would've gotten used to aliens and stopped caring, making Roger try to become the hot new story with schemes. Also, Klaus would've become a more mobile animal in Finances With Wolves, but didn't for some reason.
In "1600 Candles" Klaus spends the episode in a hamster ball though. "Check me out! I'm MOBILE!!"
Klaus was originally French but Dee Bradley Baker didn't know French well, so they retooled the character to make him German.
Steve also had a different design (read: he was really skinny and awkward-looking), and was going to voiced by Family Guy writer Ricky Blitt. This design can be seen on an early trailer for the show on the Family Guy: Freakin' Sweet Collection DVD.