if can I use Teensy 3.1 OctoWS2811 Adaptor with LPD8806 IC?
No, not really in any way that makes sense.
You could use pairs of the signals to control the clock and data pins on LPD8806, but since none of the pins are to the SPI port DOUT or SCLK pins, you'd be stuck with bit bashing. That's probably not very useful.