It's likely a variation on the idiom "take up the pen," which means to start writing.
Here's how it works:
* Quill: A traditional writing tool made from a bird's feather.
* Take up/Pick up: To begin, start, or engage in something.
* The pen: A more general term for any writing implement.
Example:
"After years of silence, she finally decided to take up the pen and write her memoirs."
So, "pick up a quill" would mean to start writing.