The most common and most likely answer to this crossword clue is the 8 letter word SHARPISH
I've seen this crossword clue in The Times
S
H
A
R
P
I
S
H
Buddy explains!

The straight part
We know that "Quickly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quickly = sharpish.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 SHARPISH mean?
SHARPISH adjective- Somewhat sharp (sudden, abrupt).
- Somewhat sharp; not blunt.
- Rapidly, soon.