It seems that what's going on is that the first LED in the strip is picking up induction from the sharp edges of the clock signal, just enough to light the green LED a little. This induction rounds off the edges of the clock (removing the high frequencies) so that the LEDs farther down the strip don't experience the phenomenon.
With my strips, the first LED does not light green when I apply power & ground, but it does light when I apply clock with no data. I'm clocking at high rates, using the SPI library.
Myself, I don't find this to be a problem. The green LED is quite dim, and doesn't interfere with light patterns that I've programmed into the strip. In fact, I use it as a handy monitor, so I can see my strip is getting clock.
If you want a fix, round off the clock edges. A small RC circuit will do this, as shown by other people on this forum who've used either a capacitor or a resistor to solve the problem. (There's probably enough trace resistance and capacitance in the strip that either one would do.)
Food Processor with Ears