The most common and most likely answer to this crossword clue is the 10 letter word POST-HASTE
I've seen this crossword clue in Independent
P
O
S
T
-
H
A
S
T
E
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 = post-haste.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.
Crossword Answer definitions
POST-HASTE noun- A great hurry.
- Quickly, as fast as someone travelling post; with great speed