I tried to start my Honda civic 1992 today but it doesn't turn on. My car was working very well last night and I didn't have any problem to start and turn on the car. Today was very cold about 12 degree. The battery is good and I can get lights and power. I got the distributor replaced 3 month ago and I also replace the belts.

I suspect you need to remove and fully charge or replace the battery. Most auto parts stores will test and charge batteries for free.

Does the Check Engine Light come on and go off after two seconds, when the ignition switch is turned to ON (Position II)?

Like I mentioned the battery is working well. I can start the car very good but doesn't turn on. The check engine light is not on.

Perform tests in the link below:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-selected-other-vehicles-46563-11.html#post2123675