"Call an elevator" is almost non-existent today except in very posh hotels where there are still people whose job is to run the elevator for the guests. In this case, you "call an elevator". Today, you do it yourself for convenience etc. So you just push the button for it. If you want someone to push the button for you, you merely say "Can you push the XXX floor button for me?". Asking someone to "call" it for you is not good because that person thinks you are demeaning him as an elevator operator.

If you want someone to push the button so that elevator comes to your floor, you should say "Call the elevator"

You can ask someone to HAIL an elevator, or more informally CALL UP an elevator, which implies pushing the button and waiting for it to come up to you.