My 99 honda accord won't start and I was told that a relay under the passenger side dash was going out, could it be that it went all the way out?

Well does the alternator start. If so than no but if yes it could be and check also the fuses.

Does the Check Engine Light come on and go off after 2 seconds, when the ignition switch is turned to ON (Position II)?

Since you didn't respond, follow tests in Sections A and B below:

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

The altenator and starter and battery are good, its seems like nothing is getting a signal to do its part and there is the click coming from under the passenger side dash and that's it

Well than check the relay. I don't know much but I think that click means the relay is trying to make contact. Its pretty simple stuff I resume toi check a relay not so simple though to get at it.

Did you check the fuses yet?

I really think and HOPE that one of the contacts between the alternator and the relay is not in contact.

Turn the car completely off and if your not sure its off take off the battery. This is done not to fry the multimeter!

First check the relay and wiring to alternator. Doing a continuity test (using resistance) using a multimeter at the appropriate range of ohms. If you didn't find any faults check the alternator contacts and if there is no continuity at all it might be that the brushes inside the motor are worn out.

You can replace them by buying new ones but it will be a lot of work to open the alternator.