In most countries hunting was a royal and noble privilege until the modern revolutions. So any limits on hunting would be self-imposed by this nobility. I know the English nobility limited its hunt on foxes rather early, though I don't have an immediate source on the date. Though I know it should be in Keith Thomas' Man and the Natural World, which I have so if you want I can look it up.

An early universal restriction on hunting came in the "Three Leagues" or Drei Bünde in what is now Eastern Switzerland. The protection of the alpine ibexes came already in the early 16th century, as it was regarded a symbol of their state. Even royals were theoretically banned from hunting ibexes, the penalty for repeat offense of which was death. I imagine that hunting seasons were also put in place there around the same time.

But in most states with royalty and nobility the hunt was probably year round if the king wanted it to be so. Hunting rights were a serious concern in the French revolution, as any property damage or even casualty that a landowner or tenant incurred due to a noble hunting party coming over his land was his own problem.