How can i add an LED for identify working status?
I recommend lighting the LED using the STM32 since you can determine if a valid lock has been made once you deconstruct the GPS module's messages.

One other possibility:
Some GPS modules allow you to configure them to only output the 1 Hz PPS if a valid connection has been made. If you can configure the module to do this, then you can connect an LED to the 1 Hz PPS signal and it will flash at 1 Hz if a GPS fix has been made.  

Note: The default configuration of most modules will simply output the 1 Hz PPS regardless of GPS fix status.