According to some information I read, a Summer lasts 16 days in Don't Starve Together.
According to the Wikia page of the Antlion you linked to:
When the Antlion first spawns in Summer, players have about 4 days to pay tribute before it becomes enraged. [...] Players can either contribute Rocks, Trinkets, or Eggs to the Antlion and it will take away a bit of its rage. [...] One should keep in mind that its rage timer cannot exceed 6 days, so players may need to contribute a hefty amount of items to it at least twice throughout Summer in order to never be attacked.
So you can't just feed it tons of rocks on the first day and hope that it will leave you alone for the entire Summer. If you want it to leave you alone for the entire Summer, you will have to give it a frozen Heat Stone (put it in the fridge), give it to the Antlion and then kill it (it might be difficult if you're not used to it though).
You need the special Summer helmet to be able to move through the sand storm though. You can fish it up in the oasis during Summer. But if you're not lucky, the Antlion might get enraged before you even find it.
Alternatively, as the Wikia says, you can feed it twice in a Summer. Since a rock gives 0.33 days of respite, and the maximum is 6 days, then 3 rocks means around 1 day, so 18-19 rocks for 6 days.
Which means:
- First 3 days, you're fine.
- During the 4th day (before it ends), give it 18 rocks (or 19 if you're unsure).
- Come back 5 days and a half later to be safe (total: 8-9 days) and give 18 or 19 rocks again.
- Come back 5 days and a half later (total: 14-16 days). If Summer isn't over yet, you can give enough stone to last until Summer is over, just to be safe.
Personally, I usually kill it when I can, because you also earn some rewards for dealing with it:
If killed, the Antlion drops 4 Meat, 6–8 Desert Stones, a Lazy Deserter Blueprint, 2–4 Rocks and varying amounts of Trinkets.
The trinkets can then be brought back to the Pig King for gold, or to the Antlion on subsequent Summer.