Should both lights in the ecu go out when the engine is running

Welcome to NICO Club!

Which year engine and ECU are you using? Which light (CEL or on the ECU and what color(s) please)? Blinking or just on? That will be helpful info, thanks.

If OBD-1, the computer could be stuck in the wrong mode and you'll need to use the key and turn the screw to set it back. Then again, it could be that they should be on, or off. I'm not sure. I would look at the FSM for that particular engine/ ECU combination. You might find yours here if you look around:
http://www.nicoclub.com/nissan-service-manuals

It's the red top nics 1989- not sure of the year, I get both lights when the ignition is switched on, but the two lights red and green goes out when it starts, now my engine only turns over and won't start

I think that is proper function for the ECU lights. Look for the instruction to pull codes by switching the ECU modes. Try to put the ECU in Mode 1 to be sure. I think you turn the ecu screw all the way counterclockwise, then put in the key and turn it on but not start, then turn the ecu screw all the way clockwise until it blinks the lights, when they are blinked just once, turn it back counter clockwise all the way and turn the key off. The different blinks are for the different modes 1-4 I think. I don't have one and this is my understanding, check out the instructions by others to make sure you're doing it right. Then you can proceed to diagnose your no start conditions.

Thank you, will try that in the morning

The engine is from a R31, made some checks and found out. I have done it and the engine still doesn't run, I got two codes today, 21 and 22, one is the ignition signal and the other is the fuel pump circuit, also got 44 after I did some checks which is telling me all is ok. I have checked my sparks plugs and cleaned them and they are now not sparking, what could be the problem?