“Quid pro quo” (something in exchange for something else) has nothing to do with calamari (squid), but there are jokes. “It could be the squid pro quo for a bowl of cold spaghetti in a horror house” was printed in the Minneapolis (MN) Tribune on May 28, 1981. “I knew when my squid pro quo was being one-upped” was printed in The Sentinel (Santa Cruz, CA) on August 20, 1981.
 
“Calamari: squid pro quo” was printed in the Ottawa (ON) Citizen on January 8, 1988. “Squid pro quo” was entered in the Urban Dictionary on September 19, 2003. “If I offer to do something for you in exchange for calamari, that’s known as a squid pro quo” was posted on Reddit—Oneliners on February 20, 2020.
   
       
Wikipedia: Quid pro quo
Quid pro quo (“something for something” in Latin) is a Latin phrase used in English to mean an exchange of goods or services, in which one transfer is contingent upon the other; “a favor for a favor”. Phrases with similar meanings include: “give and take”, “tit for tat”, “you scratch my back, and I’ll scratch yours”, and “one hand washes the other”.
 
Wikipedia: Squid as food
Squid is eaten in many cuisines; in English, the culinary name calamari is often used for squid dishes.
           
Newspapers.com
28 May 1981, Minneapolis (MN) Tribune, “Squid hasn’t been getting the kind of ink it deserves” by Kate Perry, pg. 1C, col. 3:
It suffers from squiggliness in its uncooked form. It could be the squid pro quo for a bowl of cold spaghetti in a horror house.
 
Newspapers.com
20 August 1981, The Sentinel (Santa Cruz, CA), “Wally Trabing’s Mostly About People: I Squid You Not,” pg. 23, col. 1:
I knew when my squid pro quo was being one-upped. 
   
24 July 1985, Newsday (Long Island, NY), pg. 6, col. 1:
International Squid Pro Quo
By Laura Durkin
 
Newspapers.com
2 February 1986, Boston (MA) Globe, magazine, pg. 53, col. 1:
Squid pro quo
By Robert Levey
 
Newspapers.com
8 January 1988, Ottawa (ON) Citizen, “Ins and Outs for 1988” by Nancy Gall, pg. B1, col. 4:
Calamari: squid pro quo
   
Google Books
Mortal Sin
By Paul Levine
New York, NY: HarperCollins Publishers
1995
Pg. 71:
Fried calamari was the judge’s entree of choice, which prompted local columnist Carl Hiaasen to wonder if there were a squid pro quo
 
Google Groups: alt.humor.puns
Puns of the Weak 8/3/01
Stan Kegel
8/3/01
(...)
“Let me have a bite of your calamari and I’ll pay for the whole thing.”
“Are you suggesting a squid pro quo?” (MacNully)
     
Urban Dictionary
squid pro quo
A thriving barter system in very isolated communities, involving the exchange of mollusks for other goods.
That exchange was totally squid pro quo, I gave him some squid and he gave me some tentacle videos.
by Kuroko September 19, 2003
 
Google Groups: alt.humor.puns
Kids Puns of the weak 07-27-04
Stan Kegel
7/27/04
(...)
An agreement between a restaurant owner and a relative of the octopus.  The creature would provide ink to print the menus and the owner would take calamari off the menu.  It was a squid pro quo. (Non Sequitur: Wiley)
     
Google Books
Double Dealing III:
The Pun
By Chuck Closson
Xlibris Corporation (Xlibris.com)
2013
Pg. 179:
Your calamari and my octopus is squid pro quo.
 
Twitter
The Cultured Ruffian
@CulturedRuffian
Why don’t they just call calamari appetizers a ‘squid pro quo’?
5:18 PM · Sep 15, 2015·Twitter for iPhone
 
Twitter
Megan Batoon
@meganbatoon
If a restaurant gave you calamari on the house for coming, would that be squid pro quo?
2:06 PM · Jan 16, 2018·Twitter for iPhone
   
Reddit—Dadjokes
Posted by u/shahooster October 23, 2018 (The exact date may be incorrect.—ed.)
If you offer money for calamari
is that considered to be squid pro quo?
 
Twitter
Simon Little
@simonplittle
Business idea: Calamari food truck called “squid-pro-quo”
2:06 PM · Nov 27, 2019·TweetDeck
 
Reddit—Oneliners
Posted byu/dmatlack1023 February 20, 2020
If I offer to do something for you in exchange for calamari, that’s known as a squid pro quo.