Salaam Sister,
I do not know any verses in the Quran nor searched for them, but there are verses that Allah says that they have communities and how they pray to him.
I see it this way, and this is just me, who are we to decide whether a cat, dog or any animal should not be able to procreate anymore? It is the owners responsibility to keep them separated if the opposite sex of the animal is also in the home, or if your pet roams outside on their own.
I truly believe deeply it is wrong to do that to an animal as they have the same rights as we do. And since they can not speak and tell you if they want one or not, or feel it is not a right of another creature to make that decision.
Population is not really an issue nor reason. Allah gave them the same rights to mate. It is like another of Allah's creation making a decision for humans to get a hysterectomy or men getting castrated, or when giving babies circumcision. Allah all his creations perfectly.
I feel humans is a population issue in this world, but no one is granted a life be it animal or human without Allah's permission. Allah is the one who decides if an offspring will be created whether from animal or human. I am an animal lover to the extreme, this is my beliefs. I feel the same about killing animals and eating them, but I also know Allah did not prohibit that, so I do not say do not do that, but I myself refuse to. Of course also, if your pet is at risk health wise, and your doing what is in the best interest of your animal, Allah knows what your motive was. Again if it is so they do not procreate, you are not the judge of that, Allah is. As there is nothing in the Quran that I have knowledge about, so just my opinions and beliefs.
I was told many times to get my dog neutered or he will get all kind of problems medically. Alhumduillah he has not and going on 13. My two female dogs I got spayed because vet told me they would die since they got pregnant and were still puppies. I thought I did what was in the best interest of my dogs life. Soon after that they had so many medical issues and complications, passing away earlier that what usually most dogs do. I regret I did not research it before I believed the vet, I was blind and ignorant and believed.
It is all predictions vets make. Animals life span is shortened and they get complications if they are spayed or neutered from studies that were done.
I truly believe deeply it is wrong... You will start having medical issues with your pet as well and they will not be the same.
Just my opinion and beliefs..
Brother Joseph might have something out of the Quran that is an overall message from Allah that covers this do or don't of this matter or similar.
Salaam