Can I manually turn on and off the interrupt timers?

Can I manually control(on and off) the interrupt timers of an Atmega328p chip with a push button? If yes, can it be used in compare match mode or input capture? If someone post some examples of it then it will be much helpful thanks in advance.

You can. If you want to work with the Hardware Timers then IMHO you need to study the Atmega 328 datasheet carefully.

...R

Yes: have the Arduino read the button press normally, and set the timers on/off based on it.

You'll be able to find lots of examples on how to use the timers, but indeed having the data sheet at hand helps a lot.