How do I connect 1/4" plastic tube to 3/8" plastic tube?
Watts connector described has a push-in connector for poly tubing. What size is the push-in connector for? You cut off the poly with a sharp knife and push it into the fitting if it's the proper size.



Other options, Universal Dishwasher drain coupler. Usually, they come with a new disposer. They look something like this. If you search online, all sorts of sizes are available. 



Also, given that 1/4" tubing is a standard size, you can adapt it up in size to 3/8" by using an appropriate brass compression fitting. I'm assuming that it is hard poly and barb fittings won't work as it isn't flexible enough. The compression fitting ring will have bite into its surface enough to seal and hold. For Vinyl and Poly type tubing, they make a brass insert so the compressing fitting has better bite. Shown is a sample of a union coupler with inserts, what you need will have to be selected for your situation