First of all, If you want to catch bluejays, you are going to need to have the scent compass (I believe that's what it is called) unlocked. The scent compass will point colored arrows to your prey (blue for bluejay) and you can go the direction of the arrow to find your prey.

(I'd also recommend having the active skill "Study Prey", but that is optional.)

Once you are prepared, go to the Fairywood, or a different location that bluejays spawn at. Crouch near one of the sides of the Fairywood, and wait for your scent compass to tell you what prey it is.

Ignore green/white arrows, as those will be doves and frogs. However, a light green arrow is a turtle, so you can go ahead and get that. Blue arrows are obviously bluejays, which are what you are looking for.

Anyways, if the arrow is not the color for what you want, exit and re-enter the Fairywood (or another location with bluejays) and wait for the arrow to tell you what it is, again. Repeat this until a blue arrow appears.

When the blue arrow appears, go to where it points to. If you got the "Study Prey" active skill, then use it. If you miss the bluejay, it's okay. You can always keep repeating this process. When I did this, I kept missing the bluejays, but eventually you will get one.

(Hopefully you guys are better hunters than me!)

This strategy should work for other prey too, you just have to make sure you do it at a location they spawn in, and that you know the arrow color.

This was my first guide, and I really hope it helps!