So don't call Serial.begin(). The Serial pins will be left untouched as floating inputs and the LEDs will be off.
For the LEDs showing the comms with the bluetooth, you can unsolder them or it's usually easier to unsolder their associated resistors. If you don't have a soldering iron and you can't wait for your birthday to get one, then just destroy the LEDs with needle-nose pliers. Crush them, rip them off the board.