There are some good exercise which will certainly help you to see the results in few weeks if you do them regularly few times a week, but you should always know that the most important part of the jump training is warming up and resting.
First of all you should understand that every muscle has a limit, but under a proper training you can maximize the potential of a certain muscle to achieve better results.
Muscles that you need to train are your calf muscles and quad muscles in your legs, which are the most important muscles at jumping! As said before, you should always start a workout with warming up.
It is very good to help yourself with jump rope, or 1mile jogging.
After you are done with that, start with exercises (if you are older you can also include some weight in your exercise) like box jumps, skipping, 180° jumping and others that will make your muscles stronger and more explosive! Be sure to stretch after you are done with your training.
These are just some basics that will slowly help you to improve vertical.
A lot of people also use the method of being on their toes all the time.
This exercise will not take you any time, all you have to do is to stand on toes, walk and run on toes.
This will increase the strength of your muscle and improve your vertical a lot.
You shouldn't expect huge results soon when trying to increase vertical leap.
You should know that each centimeter is worth hours of training and you should be happy with any progress.