Help me to Identify whats wrong with this Circuit
A. With no power input and a battery voltage greater than (11.0V + 0.6 Vbe + 1.? Vled), current will flow through D6 and the base of Q1 to light the red LED.

B. With power connected and a charged battery the same current path will turn on Q1 and light the red LED.  The output of the LM317 is about 1.25V + Q1 Vsat greater than the Vf of the red LED and so will always be high enough to turn on the green led. In other words Q1 turns on when the battery is charged but does not drop the LM317 output voltage low enough to prevent the green LED from turning on. 

You should disconnect the battery when you disconnect the power.

A simple fix to make the green LED turn off when the battery is full: