Your not going to eating the wood are you? PT would hold up better to the elements and is typically used in outdoor applications. You could always use EMT conduit or PVC. Both typically cost around the same and won't warp either. EMT would be better for the outdoor conditions though.
If your just asking about the boxes, then I'd use PT. Not sure if you meant for the actual trellis part or not.
Just for the boxing only. I do have cedar support poles & framing. Its just that 1 x cedar is so darn expensive. Chop & Brew site used a non treated wood. Looked like standard pine for their boxes.