"writing quickly" is correct and usable in written English.
You can use this phrase when you want to emphasize speed in the act of writing (e.g. I need to finish writing quickly).
This most accurately reflects a real use case, since users typically fix all typographical mistakes in preference for writing quickly.
But concerned citizens (hate that anodyne term, but I'm writing quickly here) shouldn't comfort themselves that it's all been ginned up.
It was quite a fashionable idea in the 70s that by writing quickly and without corrections you could avoid the falsifying input of the intellect.
Her writing quickly became the quintessential commentary to the 1960s, which was unfolding, raucously, and sometimes violently, around her in New York.
His left hand began very slowly to form the letters "P" and "I"; but then, as though taken over by a ghost, the hand suddenly began writing quickly and fluently, crossed out the "I" and completed the word "PENCIL".
Dickens tended to use rough blue paper, not unlike the blue paper used on Warren's Blacking pots, which he'd tear in half, writing quickly and firmly on one side only, sometimes using the reverse for corrections and additions.
The sketches were written quickly.
Then write, quickly, something succinct and interesting.
He wrote quickly, without the false starts that were typical of his writing.
To get a good score, a student must read and write quickly.
They stay on campus for academic breaks, though their resolve to write quickly fades.
