The most common and most likely answer to this crossword clue is the 4 letter word SPAT
I've seen this crossword clue in The Daily Telegraph
S
P
A
T
More clues leading to the result SPAT
- Second man or woman joining in quarrel
- Gaiter disagreement
- Row leads to some peevishness at table
- Expelled for minor disagreement
- Father interrupts street quarrel
- Dispute on chairs around piano
- Splash / short gaiter
- Bugs turned up, provoking petty quarrel
- Pants on fire, missing point of argument
- Father goes after son, having tense disagreement
Buddy explains!

The straight part
We know that "Quarrel" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quarrel = spat.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SPAT mean?
SPAT verb- To evacuate (saliva or another substance) from the mouth, etc.
- To emit or expel in a manner similar to evacuating saliva from the mouth; specifically, to rain or snow slightly.
- To utter (something) violently.
- (hip-hop) To rap, to utter.
- To make a spitting sound, like an angry cat.
- The spawn of shellfish, especially oysters and similar molluscs.
- A juvenile shellfish which has attached to a hard surface.
- To spawn. Used of shellfish as above.
- (often in the plural) A covering or decorative covering worn over a shoe.
- A piece of bodywork that covers the upper portions of the rear tyres of a car.
- A brief argument, falling out, quarrel.
- To quarrel or argue briefly.
- A light blow with something flat.
- To strike with a spattering sound.
- To slap, as with the open hand; to clap together, as the hands.
- An obsolete unit of distance in astronomy (symbol S), equal to one billion kilometres.