Think about it in very simple terms. Foreign policies and geostrategy are like chess. Every move you make has implications. Think in terms of possible scenarios. The given is very simple and clear: China has a quarrels with India that goes decades back, (but India was traditionally quite friendly with Japan and USA), this is a fact.
Now, even if India was NOT in a Quad alliance with the other three, she will still take advantage of what they perceive as a dire situation for China. And China would be in a dilemma to try to let it go without provoking a war. This is because China want to focus on US and her two Allies. Starting a war with India will be diverting resources that doesn't serve the cause of countering the US, and at the same time ruin China's reputation. If US (and the other two) provokes, the only effective and proper counter would have to be right back at the US (and the other two).
However, If India goes into an alliance (Quad) with the US and the other two. This changes everything, because the US (and the other two) now has an obligation to fulfill the duty of the alliance agreement. This means that India would now become a liability of the US. Because after the formation of such alliance, China will have the extra option of attacking India as a way of retaliating against any provocation by the US (and the other two). What's worse, is that the US will have to divert resources to help India, if it falls into a bad situation with China. Should the Indian army not be able to withstand the attack of the PLAGF, the US and her other two allies/servants will have to sail far away from home into the Indian Ocean to fight PLAGF (think about the logistics and where the sea route would be located).
From China's Point of View, the Quad is the most desirable scenario, because once the Quad forms, all of China's military asset would become strategically highly useful again. PLAGF will no longer just be sitting at home watching the Navy and Air force duking it out with the USN, JP and AUS. PLAGF can join the fight too, AND it won't be on the motherland itself, AND it will be in a situation where their supply lines are much shorter and secure than that of the US, Japan ad Australia.
Of course, the US, Australia, and Japan could simply refuse to actually fulfill their alliance obligations. China wants to drag them into a land war on Indian soil, they could be cunny and refuse to take the bait. And so, they will ignore Indian's plights, and basically ruin their own reputations. When India lost hope in the other three, China will be swift to cease fire and steer Indian mindset to China's strategic intentions.
