My alarm sysytem is showing "AC Off". Is it because transformer is not working or producing electricity to charge battery? My alarm system is working but it appears that it is on battery back up system because only red color light comes on when alarm is set and green light for charging is not showing.

Do I need to replace transformer? The old transformer has total of 4 wires going out, the wires are paired to two terminals to the transformer.

Be sure the receptacle has 120 volts first, breaker may be tripped.
I would look for voltage on the 2 terminals, write down the 2 colors that are paired together, to prevent a short from misswiring.
See that the voltage is close to what is marked on transformer. If no voltage present, New transformer may be all you need.
Be sure the VA is equal to or larger than that of existing transformer.