The Quadrilateral's prime purpose is to contain China's strategic expansion in the relatively newly labelled region "Indo-Pacific" (see map below). Within the "Indo" (Indian Ocean) Chinese surface ship and submarine visits have risen sharply in the last 5 years. China has also established a naval base at Djibouti, Africa, facing the Indian Ocean.
Within the Pacific China is of course building militarised South China Sea islands and bilaterally courting or pressuring, some Southeast Asian nations (Vietnam, Philippines, Thailand, Cambodia, Laos, Indonesia and Malaysia). China may argue that this activity is all part of its oil-gas trade route One Belt One Road Initiative but the raw uncompromising militaristic flavour of China's advance demands containment by other Indo-Pacific countries.
The US, Japanese, Indian and Australian members of the Quadrilateral will most notably interact in naval exercises, but also quiet diplomatic meetings continue between them.
If Trump had not dumped the TPP negotiations the Quadrilateral countries might have also formed a powerful economic grouping within the TPP. Instead the Chinese led Asian Infrastructure Investment Bank (AIIB) is proving successful while the TPP proposal collapses.
To make the Quadrilateral work Reality Twitter Trump needs to foster consistent US leadership.
As in 2007-2008 China may yet flex its economic muscles to break the Quadrilateral (the weakest link unfortunatly being Australia).
The Indo-Pacific Region, an expanse that well fits containment of China by Quadrilateral nations: Japan, India, Australia and the US.