This is just a slightly special case of the general question, "How could we tell if an alien race is intelligent or convince an alien race that we are intelligent?" Adding the detail that they are giant ants and the human ship has made an emergency landing would likely change the details, but the big question is tough enough without getting into details. I'm sure thousands of science fiction stories have been written exploring this idea.
For starters, what is the definition of "intelligent"? We often talk about using tools. But people have trained animals to use simple tools. I saw a documentary years ago about animals in nature using simple tools, like a bird trying to reach the water in a bottle with an opening too small for its head, dropping pebbles into the bottle until the water level rose to where it could drink it. Was that "intelligent"? Etc.
While the aliens in most SF stories think and act just like people, the only difference being that they have blue skin or pointy ears, some writers have tried to craft a truly alien intelligence. (Aside from the dull standby "they have no emotions", which always leaves me wondering, Then why do they do anything at all? It's a paradox to say that they have no emotions because they are so proud of their refined logic. Isn't "pride" an emotion? If they don't love or hate anything, then that must mean they don't love life. So why would they lift a finger to protect themselves from a wild animal attack or a heavy object falling on them? Wouldn't they all be dead within a few months? Etc. But anyway ...) Could aliens have a totally different mathematics, for example? Could they never have thought of the idea that 2+2=4 but be very advanced in, say, Euclidean geometry? It seems theoretically possible but difficult to imagine how it would work in practice.
Would aliens wear clothes, or is that something peculiar to humans? One could imagine aliens who do not get sexually excited at the sight of each others' bodies -- maybe who don't reproduce by sex at all -- and who have never thought of the idea of fashion, i.e. clothes for decoration, and so only wear clothes when necessary to protect them from extreme weather, etc.
Aliens might communicate in a way imperceptible to us, like they might have radio transmitter and receivers built into their bodies. Or they might communicate in a way that we can perceive, but that would be too far from our thinking or too subtle for humans to notice. Like emitting odors in a specific pattern, blinking their eyes in Morse-like code, etc. I'm reminded of an SF story I read many years ago -- sorry, I forget the title and author -- but there's a scene where humans meet aliens, and eventually one of the humans figures out that the aliens communicate by sending microwave messages. At this point one of the other humans says, "You mean they communicate by telepathy?" And the first person says, "Sort of. But to them, so do we. They don't think of sound as a method of communication, so when we talk to each other" etc.
Anyway, books can and have been written on this subject. I can't imagine you'll get a complete, definitive answer here. I'm just throwing a few thoughts in to chat, basically.