You can catch Walleye in Stardew Valley primarily during the Fall season on rainy days, between 12 PM and 2 AM. The best and most common locations to catch Walleye are:

• The pond in Cindersap Forest, which offers about a 32% chance of catching Walleye on rainy Fall days.
• The river in Pelican Town and the river in Cindersap Forest.
• The Mountain Lake.
• The Forest Farm or Riverland Farm pond, if you chose those farm types.

In Winter, you can also catch Walleye if you use a Rain Totem to create rainy conditions. Additionally, Walleye can be found randomly in garbage cans during Fall and Winter, or purchased from the Traveling Cart. A pet cat with max friendship can also gift you a Walleye