"Near" Knoxville depends on how one defines "near".
I usually drive over 4 hours to find my nearest good turkey hunting.
Your best bet is to luck into knowing someone who has turkeys on their land, doesn't hunt, but will let you hunt. This can be a low odds or time-consumptive endeavor.
There is a good bit of public land around 1 to 2 hours away, although some of it provides only very limited opportunities to hunt via quota draw hunts.
The Catoosa WMA has a respectable turkey population, and a fair amount of hunting opportunity. It's not as good as a lot of private properties, but it's not horrible. It's a beautiful place. The Chuck Swan WMA may actually be better than many private properties, but your opportunity to hunt there is very limited, maybe one weekend a year, maybe.
Like I said, I drive over 4 hours to find my nearest good turkey hunting, albeit many people around Knoxville have much better hunting much closer than do I. Heck, my turkey hunting hasn't even been that good the past couple years, but I'm a creature of habit. If I were looking for something different, closer to Knoxville, where I could hunt a lot, I'd likely be spending much time on the Cherokee National Forest. "Tough" terrain hunting, but "real" hunting, and not as much competition from other hunters as you may find on many other WMAs.