I'm disgusted at you people. How dare you even consider that His Holiness, Jim Davis, would create anything but wholesome, family friendly, cat-centric humor? Like that series of strips in which it turns out Garfield lives in an abandoned house, and is driven insane by his loneliness. Jon and Odie are hallucinations concocted by his love-starved decaying mind. Family. Friendly. Fun.
So no, Jon didn't gulp down a healthy swig of dog semen, you disgusting perverts. He merely gulped down a healthy swig of dog fertility drug. There's nothing more family friendly than fertility drugs, that's how we get families in the first place! You may ask why there's a piping hot cup of dog fertility drug sitting next to Garfield in the vet's office. You may also ask why Jon would blindly pick up a cup of fluid at a vet's office and shotgun it. Those are valid questions, and ones easily answered if you know a bit of Garfield lore. Firstly, according to canon: Jon is an idiot. He's a slack-jawed barely functional adult who, after years of dealing with Garfield's abuse, has had the last meager ember of his sanity splutter out. He will drink any mystery fluid he finds, no matter the context. The world and all the things in it are a constant mystery to him. He bumbles around, trying doors and picking things off the ground and sticking them in his mouth, naive as a newborn child.
So there's your answer to that question. But why would there be dog fertility drugs next to Garfield? Well, as we've already established, the entire Garfield universe (The Garfyverse) is a figment of an insane cat's atrophying brain. As the neural connections in his prefrontal cortex begin to tangle and become withered, Garfield's ability to reason diminishes. "I am cat," he thinks sluggishly. "Cat go to vet. Vet make the puppies. Drink the puppy drink. Jon idiot. Hate Mondays... hate." Shhh, sleep, Garfield. Sleep. Let the blessed darkness take you.