The Quartet does not command any military power, it cannot directly affect the situation on the ground, what Israel likes to call "facts on the ground". It cannot directly compell Israel to follow internatioinal law and the Geneva Conventions, nor the stipulations of previous peace agreements Israel has signed like the Egypt peace agreement, Oslo, Wye River Accord, or the Road Map. It might be meaningful if the Quartet could override a US veto at the UNSC.
US rally shows support for ex-Marine held by Iran on spying charges (AP)
from the article: Abbas: Britain and France would recognize Palestinian state