Hi, I have LCD Nokia 5110 which is working perfect on Arduino UNO but if I connect it to Arduino Nano on same pin numbers, it does not show anything. I have tried using different pins but no output.
I'm using Adafruit_Gfx library with pin numbers defined in the library. I also tested the example code given but can't get any output.
I am presuming you are powering it from the 3.3 V pin on the UNO or Nano.
Depending on what sort of "Nano" it is, the 3.3 V is derived not from a dedicate regulator chip as on the UNO, but from a FT232 or CH340 chip which cannot supply sufficient current for the display. Try a separate 3.3 V power supply (or if you cannot find that, just try the UNO 3.3 V when operating from the Nano). 
- make a clear picture of your setup where we can see each modul and each single wire and post it to the forum.
- post your sketch
Connections are same for both Arduino UNO and Nano.
RST --> Digital pin 9
CE --> Digital pin 10
DC --> Digital pin 8
DIN --> Digital pin 11
CLK --> Digital pin 13
Output is attached for both boards. I used 5V without any resistors on both boards.
Arduino Nano is Atmega328P (Old Bootloader type).
If you need more info, I can provide.
If this is the best picture you have I can't help because I can't identify where which cable is going, nor how your display and your nano is powered, nor if it fits to your sketch. May be someone else...
Nano and Uno use the same pins so, if the wiring and code is the same, it should work. There may be a power problem, so try disconnecting the backlight. These divices are just seconds, and the problem may be mechanical. Try pressing lightly on the screen. Also, check the breadboard connections are kosher.