It's absolutely the quickest way to transfer at Green Park. Not only is there less distance to cover, most of it is covered for you, mechanically.

The TfL-suggested route may be quicker, despite the two changes. Piccadilly to District at Hammersmith is a cross-platform change (if it's quiet, I'd suggest Barons Court instead, as it's a rather lovely station), and District to Jubilee at Westminster can be done on an escalator or lift without much walking.