Have never seen anything specifically relating to a climbing turn.
But I assume that you've got access to, and understanding, the forces in a turn, and the forces in a climb? If so, you can just combine these together.
The classic "forces in a turn" diagram shows that, in the vertical plane, the vertical component of lift is reduced, and a horizontal component of lift is created. That is true in a climbing turn.
The classic "forces in a climb" diagram shows how the lift, thrust, drag and weight all interact. Simply replace lift in this diagram with the vertical component of lift.
I'd guess that an aerodynamics expert will find fault with my simple explaination, but it should be good enough for most uses, so I hope it helps.
FFF
--------------