The most common and most likely answer to this crossword clue is the 7 letter word APTNESS
I've seen this crossword clue in The Sunday Telegraph
A
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More clues leading to the result APTNESS
Buddy explains!

The straight part
We know that "Fitting quality" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fitting quality = aptness.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 APTNESS mean?
APTNESS noun- Suitability; the quality of being apt or suitable
- Disposition of the mind; propensity
- Speed or readiness in learning; docility
- Proneness; tendency