I have a source that can deliver 20mA and 4.5V (MAXIMUM). I need to boost the current to 1 A without using any further source. Also, the voltage should be maintained constant at 4.5V. is that possible with emitter- follower configuration\? if not is there any other solution
Short answer: NO
Electrical power, which is the rate at which electrical energy is transferred by an electric circuit is given by \$P = V*I\$.
The law of conservation of energy states that the total energy of an isolated system stays constant. Now you want to have an input power of 90mW and an output power of 4.5W which is clearly not possible.
Input current 20mA, output current 1A, at the same voltage, that's a power ratio of 50:1. If you are happy with a small output duty cycle, you could achieve it with energy storage.
The size of energy store depends on the length of pulse you need. If you want a 1s pulse every minute, then that's quite a large capacitor, but a very small battery indeed. For 1mS every 100mS, a small capacitor would do. For one day every two months, a large battery.