Does the English Pokemon green for the Game Boy have the same Lavender Town theme as the Japanese version?

Pokémon Green was never released outside of Japan, assuming you're referring to the original Game Boy game. As such there's no English Green version except for LeafGreen. Only Pokémon Red and Blue were released in the US and Europe, though these versions have the 'fixed' variant of the Lavendar Town theme. The same applies to the final releases of Red and Green in Japan, though I presume the remaining first releases of those games retain the original theme if you're able to get a hold of them.

If you're referring to LeafGreen and FireRed, no, the theme is very much the same as it was in Red and Blue, albeit recomposed to suit the Game Boy Advance's sound chip.