IMO, the only way they will associate turkey sounds with danger would be if you alerted a hen and she started "putting". I do believe deer hone in on "natural" danger sounds from other wildlife.
Several years ago, I was still hunting down a creek when I spotted a nice buck about 150 yards out front, just laying on a hillside. At the same time, a squirrel picked up my movement and started "barking" at me. The buck payed no attention to the squirrel until the dang thing wouldn't quit barking. After about 15 min., their was a noticeable difference in how the buck was reacting to this barking. He stood up and was all eyes and ears in my direction. I had the wind in my favor so I was ok. He watched intently in my direction until finally the squirrel quite barking. A few minutes after that the buck layed back down. It was fascinating to watch! Don't think deer don't pay attention to natural warning signs!
Squirrel barks, turkey putts, heron squaks, etc.., if short, you're probably ok, but if persistent they will investigate.