Is there an alternative method beneath surgery?
Yes. There are treatments for the female cat, to avoid her heat (weekly oral medication or monthly injection) and for the male cat, to decrease the production of sperm (implanted hormone chip for periods between 6 month and 3 years).
Is this treatment practicable for not-breeders?
Not really. It is stress to go to the vet each month for injections or to control that the oral treatment would really be eaten. Also, the treatments are expensive and not available everywhere in the world. Additionally, it strongly increases the risk of cancer!
What would an sterilisation cause:
Sterilisation means, the tubes of the reproductive organs would be cut. So the hormones would be there (heat for male and female as normal), but the breeding is not possible, because the sperm and egg will not get where they are "needed". This would cause a lots of stress in cats. Because they feel no love like us, mating in cats is a rough thing and it seems proved that pain is essential for the reproduction in cats. (Please see the related question here and here, and try to explain it to your family.)
Less stress and cat-suffering with castration:
In this option the reproductive organs would be removed. So the female cats will have no hormones to get in heat and males no need to mount. They can live in peace together and do not miss anything. There will be no kittens one needs "to get a rid of", which (as first litter in the neighbourhood) may find a nice home, but more likely will go to a shelter, or as strays to the street, or in the worst case need to be killed (There are people everywhere, who do not feel with animals as living creatures, sadly...)