The former Arsenal star faces a lengthy spell on the sidelines in the wake of the operation, according to Sky Sports News.
The Frenchman joined the La Liga outfit from Arsenal in January and has impressed in Spain, scoring his first Valencia goal in their 2-1 win against Malaga last month.
Arsenal have tweeted their support for Coquelin following the news of his injury.
"We're thinking of you, Francis," read a message on the club's official account.
Valencia are currently fourth in La Liga, in good stead to qualify for the Champions League with an 11-point lead over Sevilla in fifth.