Hello!
I agree with Bob and would say the first sentence is better: "What bird's song do you like the sound of most?" But I would like to add that "which" would be better than "what" as there is a choice between different items.
If you wanted to use the second sentence: "What bird's song sound do you like most?" then I would write it like this: "Which bird's call do you like most?" as the word "call" can be used to describe "bird's song sound".
I hope that helps!