bitter, serious, violent
family, internal, lovers'
personal, private
long-standing, old

have
pick, provoke
I don't want to pick a quarrel with her.

be involved in, become involved in, get involved in (all esp. BrE)
make up (BrE), patch up (BrE), settle

break out

quarrel about
a quarrel about money

quarrel between
a quarrel between family members

quarrel over
a quarrel over the ownership of a piece of land

quarrel with
Our quarrel is not with the people, but with their leader.

have no quarrel with sb/sth
We have no quarrel with their plans, in fact we support them.

bitterly, fiercely, violently (all esp. BrE)

about, over
Companies always quarrel about the exact terms of the agreement.
What did you two quarrel about?
We tend to quarrel over money. (esp. BrE)

with
I didn't want to quarrel with Tyler over weekend plans.
Why must you always quarrel with your sister? (esp. BrE)