I have Esteem its AC has stopped working. When I showed it to the mechanic, he said that there was some problem with the Relay and the motor of the fan/blower. When he was testing with new Relay he found that motor was sometimes starting and sometimes needed a slight push from his screwdriver to start.

Generally this type of problem in our table/ceiling fan is solved by replacing the capacitor. In this case he is asking me to replace the motor. Does blower has capacitor in it and can this problem be solved by replacing the capacitor or the motor needs to be replaced.

There isn't a capacitor in your HVAC blower.
The internal contacts are losing continuity and the motor needs replaced, as your mechanic recommended.