It seems they were a kind of Easter-Egg.

In the novels, we received a long narrative about the elephants of The Golden Company, which was left out of the show.

In George R. R. Martin’s fifth GOT novel, A Dance With Dragons, we learn of The Golden Company through a rather long-winded side story that has thankfully been cut from the HBO show lest it get even more complicated. Someone in the writer’s room clearly wasn’t happy with this decision, and so decided to throw in an elephant-sized easter egg for viewers who have also read the series so far.

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_{Cersei’s need for elephants in the Game Of Thrones premiere comes directly from the books - Metro UK}

So, it seems like this is a special Easter-Egg for those who have read the series.

Elephants are always a funny kind of weapon

Having a big weapon on your side is a terrible advantage. Elephants are a good historical example: while they don't deal a lot of damage per se, they are terrible for enemy morale. You can only have a few, and each of them won't deal a lot of damage. But if you are on the other side, that's another topic.

Gosh, an elephant charging me! Ok, so if we all keep the formation, the elephant will crush on our spear formation. Sure, there will be casualties, but we will win. The only thing we had to do is keep our formation. Hey, what is Joe doing? he can't escape the battle, can he? Whatever, he was a coward. We can keep formation without him. But, wait, Jack is also fleeing! He was the best of us. If even he thinks we are lost... But we need to keep the formation! Oh, no, some others are also running. The formation just broke up, the odds are very bad for me. Yep, better run for my life.

Even if the elephant wouldn't do that much damage, it crushes the enemy formation, and you can now deal much more damage.

Why on earth wouldn't you want such a weapon?

Sure, the enemy also have a huge weapon, but that won't ensure that the foot soldiers will keep the formation.

And it's because she doesn't have dragons that she wants a similar, smaller armed animal that can afraid the enemy.

So that we don't expect elephants

Golden company is well-known for their elephants. Harry Strickland is proud of them, and as it's the final season, we expect massive battles. This scene is also a way for showrunners to tell the spectators to don't expect any elephant. Sorry, all CGI budget was burned on dragons.

I don't think it was an easter-egg or a joke.

Cersei demanded Elephants from Euron Greyjoy, because she wanted to give him an impossible task. She would win either way, if Euron delivered or not.

Euron Greyjoy is obviously obsessed with fame, power and, for some weird reason, also with impregnating Cersei. Knowing that, Cersei is full-on using his blind quest for power in her advantage to get as much military reinforcement as possible.

I bet even an office sitter like Cersei knows that transporting elephants with ships is a pretty heavy task. But since she is not to keen on giving Euron the feeling, he did everything she ever wanted, she demanded elephants anway. Best case: "Yay, elephants for my army." Worst case: "Oh you did not fulfill my request.. guess you don't get to sit by my side then. I'll take the rest of your army anyway, btw."

This does not fully explain why Cersei let Euron sleep with her in the end, anyway. But I think she had a change of heart (figuratively, she obviously does not have a heart) when Euron returned with his army, because she knew she was already pregnant and could probably use him some more.

(Does it show how little I think of Cersei?)

Cyvasse

G.R.R Martin is a massive chess fan, even working as a chess event organiser. As a result, there are clear chess influences in A Song of Ice and Fire, the main one being Cyvasse. It is suspected by many book readers that the outcomes of "on page" games, particularly those played by Tyrion, are of a prophetic nature. Two of the most notable pieces are Dragons and Elephants. Clearly, there is a potential for symbolism here with Targaryens (Dragons) and The Golden Company (Elephants).

Cyvasse isn't shown in the Game of Thrones world, so no in-show cyvasse prophecies will be fulfilled. However, the story is based on the books so there is a very, very small chance that book readers could extrapolate on some of these prophecies on shoe horn them into the books. The one that springs to mind is the Golden Company's missing Elephants.

Essentially, this would boil down to an Easter Egg for book readers but with a little more significance than, for example, Arya's Rat Cook pie.

The scene you're talking about showed Cersei get visibly upset at the fact the Golden Company was only 20,000 strong. Her mention of elephants was more like "did you at least bring tanks to the war?" Since it's only mentioned once, and the captain quickly lets her know they don't have them, then these would seem more of something said out of spite than a real concern.