What adapter do I need to connect this shower hose to valve?
There are countless plumbing adapters made for the seemingly infinite number of combinations a person might want to connect. It's nice when we can find a single fitting that does exactly what was needed, but a home center like Home Depot carries a relatively small selection of adapters. A person can almost always make a given connection by using a combination of two or more in-stock parts. It is not viewed as wrong, improper, nor unprofessional to use a combination of whatever is available to get the job done.

It would be great if you could find a reducing coupling -- that's a part which is female threaded on both sides, but one side is threaded smaller than the other. Sometimes called a bell reducer.

If you can't find a reducer of the right size then try for a reducer where at least the large side fits (ie 1/2 IPS to fit your valve). Failing that, get an ordinary coupler. In both cases you can use a bushing to reduce the final size to match the thread of your hose.

I can't quite tell, but it looks as if the hose has a female connection. I have on occasion seen male-to-male adapters of different sizes. You might be able to find one of these that fits your hose and steps up to, or closer to, the 1/2 target size (and then finish the connection with a coupler). If not then just use a regular nipple if needed.