Usually a softbox will reduce or block UV, but it depends on the diffuser fabric. Some softbox materials are UV-blocking, while others are not.

Studio flash tubes do emit some UV, but generally not a large amount. The main practical issue mentioned is not damage so much as color: UV can affect the apparent color temperature of the light after it passes through the softbox, which may shift your results and make color consistency harder.

So using a bare tube behind a softbox may be fine, but it is not guaranteed that the softbox fabric alone will fully handle the UV. If color accuracy and consistency matter, a UV-coated tube is the safer choice unless you know your softbox diffuser material is designed to block UV.