Yesterday, while delivering her kittens, my cat kept breathing through her mouth and breathing heavily. We thought it's because she's just too tired. This is her second time. Last time, she gave birth to four but two kittens died and the other one died too, leaving one kitten behind. Now, four kittens are alive and it's the first time she feeds four kittens. Up until now, she's breathing heavily and one of her kittens has pooped. We don't have a vet here in town.
This is a summary of information originally given in comments.
If a cat keeps panting after giving birth, the most probable reason is that she has milk fever (a lack of calcium in the body). This condition weakens the mother and can be lethal to her, but it also keeps her from nursing her kittens.
If possible, please have your cat treated by a vet as soon as possible. If that is not possible, please have a look at this question:
Pregnant and nursing cats should eat either kitten food or special food for nursing cats. There are also calcium food supplements for cats (please never give a cat any medication or food supplement designed for humans without consulting your vet first, you could poison your cat). If no professional solution is available to you, you could crush egg shells or cooked bones in a mortar to a fine powder / paste and add it to her food, but you should always prefer a professional solution because they are better digestible and contain additional vital nutrient.
THIS IS A MEDICAL EMERGENCY!
According to https://www.greencrossvets.com.au/pet-library/articles-of-interest/why-is-my-cat-panting/, The only reasons why cats would be breathing through their mouth period would be because they are Hot, Stressed, or have an underlying disease.