The 22-year-old will undergo surgery on Monday and could miss the rest of the season.
Motherwell claim the player had been assaulted in Wishaw at the weekend.
"For legal reasons, the club cannot comment any further as the matter is now in the hands of the police.
"We wish Paul a speedy recovery from his injuries."
Strathclyde Police said that a man had been arrested over an incident in the club but had no record of Quinn being involved.
A police spokeswoman said: "A 39-year-old man was arrested in connection with an incident in Baker Street nightclub. A report will be sent to the procurator fiscal."