No you don't have to treat/stain the wood, and don't use treated lumber, either.
A tub that size should only be used as a "make do" (temporary), it's too small. Gators, in captivity, need aprox. 50% water/50% land mass, not just a wooden basking spot. You'll probably also need an aquarium heater to keep the water temperature up. I'm speaking from experience, I rescued a baby gator not too long ago and used a rubbermaid-type tub about the same size.
The former owner hadn't fed the baby gator at all, the whole month they had it. It was skin & bones. I had to force-feed it a rat pup because it could barely move. Took me a week and a half of experimenting with its setup before it finally got comfortable & came to life. Now its got quite a healthy body & attitude.
If you live near a bait shop (fishing) you can get more for your money by getting small shiners or large minnows. It could take a week or two for it to get settled in enough for it to start eating. They also like crickets (thrown into the water).
Below is a link to some pics of the setup I used. I used aquarium gravel to build up one side of the tub and placed a large rock on top of the gravel. I don't recommned using gravel, though. I used it because at the time, the ground was frozen and I couldn't dig up any dirt. I never fed my gator on the gravel, always in the water, and there wasn't any gravel in the bottom on the water side.
Click Here There is a link at the bottom of the page to some more pics.
Have a good one!
HH
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It is said that 1 out of every 4 people are mentally unbalanced. Think of your 3 closest friends, if they're normal, then it's you.