Carp are native to temperate zones and must be stimulated as their ancestors were by temperate zone conditions in order to grow, multiply and appear to be content in their artificial environment when kept as property beautifiers.
Of course, you need a place in which to make the fish feel at home. Being human, you will prefer that their pond or pool be clean-looking so you can enjoy their frolics around the rocks and aquaculture plants that you have provided for them. However, the fish do not always share your ideas about the ideal living conditions for them. In nature, fish are wild animals and do not want to watch humans nearly as much as humans want to watch them. They prefer hiding areas in their watery habitat. You, on the other hand, want them where you can see them.
Koi carp are the descendants of wild carp which have been bred for variations in coloring and body markings. The koi carp are generally from Japan. Thus, the living conditions you provide for the ones you keep to beautify your property should be as near the conditions in Japan as possible. Japan has four seasons, just as much of Europe and North America. As much as possible, you need to simulate conditions in the pond to the four seasons. This involves not only water temperature but hours of daylight. In Japan, daily daylight hours are fewer during the winter than during the summer.
If you can create conditions which seem natural to the fish, they are more likely to spawn and produce some fry (baby fish). Of course, you need both males and females of spawning age. When the length of daylight hours and water temperature are maintained similar to those conditions in Japan, the fish will respond and begin the reproduction processes.
Creating the four season conditions may be a challenge, especially if you are trying to grow koi carp in areas of the world which do not have the four seasons. You will need to manufacture methods to control water temperature and amount of daylight per day. This needs to be a long-term procedure as the females will need several days of the right water temperature and length of days to begin producing eggs.
If you are able to get some fry, do not be surprised at their survival rate. As in the reproductive nature of other animals, there are more eggs than will survive and more fry than will survive. Again, since we are human, we want to be able to watch the baby fish grow. However, they will definitely grow best if they are in water which we consider to be dirty water. A lot of zooplankton, tiny vegetative plant life, is ideal for young fish. This does tend to make the water appear dirty although it is healthy for the babies.
Unless you are going into the koi carp business for commercial reasons, keep your expectations rather low. Instead of counting how many offspring your carp have, consider their beauty and how happy they make you feel at feeding time when some of them will eventually come to the top of the pond and eat food from your hands.