No, not just any joke about an elephant. Elephant jokes are a very specific sort of joke: they are always silly, always presented in a Q and A format, often non-sequiturs or puns, and generally clean.
An Elephant Joke
Q:How do you make an elephant float?
A: With two scoops of ice cream, an elephant and some root beer.
Not an Elephant Joke
There was an old man in France who used to get up at the crack of dawn every day and sprinkle white powder all around his house. When his neighbor asked him what he was sprinkling, he replied that it was elephant repellant. The neighbor exclaimed "There are no elephants in France!" to which he answered "I guess it must be working then!"
Sometime in the late 1950s or early 1960s, elephant jokes started to become popular. By 1963, they were such a fad that Times magazine ran an article on them, giving examples including peanut butter, Tarzan, and colored sneaker jokes. No refrigerators, but I suspect that's only because the refrigerator was an old hat by then. They stayed big throughout the 70s, but by the 80s they were moving from the domain of the adults to the children. And now even the kids think that they're stupid, and only us nerds like them.
Q: How do you fit four elephants into a Volkswagen Beetle?
A: Two in the front, two in the back.
Q: How do you fit five elephants into a Volkswagen?
A: Two in the front, two in the back, one in the glove box.
Q: How do you know an elephant has been in your refrigerator?
A: there are footprints in the peanut butter.
Q: How do you know there are two elephants in your refrigerator?
A: You can hear giggling when the light goes out.
Q: How do you know there are three elephants in your refrigerator?
A: You can't get the door to shut.
Q: How do you know there are four elephants in your refrigerator?
A: The Volkswagen is parked out front.
Well, you see how these can snowball out of control. Chains of jokes can go on as long as you want, and can interact with other chains. In fact, the whole conflagulation of elephant jokes is so big, it contains some that don't even have any elephants in them.
Q: How do you get an elephant into the fridge?
A: 1. Open door. 2. Insert elephant. 3. Close door.
Q: How do you get a giraffe into the fridge?
A: 1. Open door. 2. Remove elephant. 3. Insert giraffe. 4. Close door.
Q: The Lion, king of the jungle, decided to have a party. He invited every animal in the jungle, but one didn't come. Which one?
A: The giraffe, because it was stuck in the fridge.
Q: Two explorers attempt to cross a crocodile-infested stream. How do they manage to get across?
A: They just wade across. The crocodiles are at the lion's party.
Okay, with the realization that some of you may actually be getting bored with these, I'll set you free; the rest of this writeup has absolutely no factual or educational content. You may leave. But since a very special sort of person finds this type of joke to be absolutely hilarious (Hi!), I'm going to include a few other joke strings.
Q. Why are elephants gray?
A. So you can tell them from plums.
Q. "What did Tarzan say when he saw the elephants come over the hill?"
A. "Here come the elephants over the hill."
Q. What did Jane say when she saw the elephants coming over the hill?
A. "Here come the plums over the hill" (Jane is color blind).
Q: What did Tarzan say when he saw the elephants with sunglasses on coming over the hill?
A: Nothing, he didn't recognize them.
Q: What did Tarzan say when he saw giraffes with sunglasses on coming over the hill?
A: "Haha! You elephants fooled me once with those disguises, but not this time!"
Q: What did the elephants say when they saw Tarzan coming over the hill?
A: Nothing, elephants can't talk.
Q: How do you shoot a blue elephant?
A: With a blue elephant gun, of course.
Q: How do you shoot a red elephant?
A: Hold his trunk shut until he turns blue, and then shoot him with the blue elephant gun.
Q: How do you shoot a yellow elephant?
A: Don't be stupid, there are no yellow elephants.