Shell-like torsos and cloacal respiration.
If what you want is simply for the animals not to appear like they're breathing, you're in luck: the method you want already exists, and it mostly involves having rigid torsos.
Birds are nice examples, but not quite. While the lack a diaphragm like mammals do, their chests also move while they breathe, as muscles in their torso move the vertebrae and sternum up and down in order for it to expand in volume.
But there's one kind of animal that, despite not having a rigid exoskeleton doesn't quite enjoy the benefits of a more flexible torso: turtles.
Turtle shells, far from some exclusively external armor like you'd see in something like an armadillo, are composed of their ribs and vertebrae. Today we believe this came as an adaptation for digging, but regardless, the final outcome is a creature with a completely rigid torso that can't really expand to let in air.
So, how do they do it? Pretty simple, they transferred the job from muscle and bone to just muscle. Turtles have sheets of muscle specialized to contract and relax in order to force air in and out of the lungs inside their shells, allowing for the lungs to expand and contract inside fully rigid, unmoving torsos.
There's also another page from the turtle book that you can use in order to make your aliens seem like they don't breathe: butt breathing.
That's right, some turtles, especially some semiaquatic and aquatic species can exchange Gases through their butts,something known as cloacal respiration.
To explain it: breathing in the water,putting it simply, mostly takes 3 things: the water containing oxygen, a place full of blood vessels to get in contact with the water and get that oxygen through diffusion and a way to make that contact happen. Fish do this through their gills, but some turtles, already having a cloaca packed with blood vessels, adapted to use said cloacas to extract oxygen from the water, especially while hibernating underwater in the winter. This greatly helps turtles with a more aquatic lifestyle spend longer periods underwater, with some species being able to get all of the oxygen they need depending on how active they are at the moment.
So overall, if you want a species that doesn't look like they breathe, make them more turtle-like. A species with a rigid, armored torso housing their lungs, as well as muscle sheets responsible for handling their breathing. Couple that with the ability to extract oxygen from the water via something that isn't as obvious as gills and you could easily sell the idea of an alien species that simply does not breathe, or that does so through some overcomplicated, highly alien method, when you're actually just dealing with space turtles.