Is it possible to transfer one app from my old phone to my new if my phone is not rooted?
Do you want to just install the specific version of the application you have in your old phone? If so, 

a) run an application like Titanium Backup on the rooted phone

b) backup the app

c) find the .apk in the backup folder 

d) transfer the .apk to your new phone

e) run the .apk to install the application on the new phone. I believe some phones do not do it out-of-the-box and you might have to download some kind of app (search for "apk install") to do it. Try without an app first, though.

In case you also want your data to be transferred along with the app, I'm not sure whether TiBu can do this for non-rooted phones. Someone else will be better suited to answer.