Unfortunately no; Using “Hey Siri” there’s no way to fully silence/turn off the beep sound. The best option is to turn the volume all the way down but that’s about it.
Strange, I don't think I even get the sound aside from when I have headphones connected (and as of iOS 11 I get this almost double activation sound with headphones).
With Hey Siri you get a beep beep after “Hey Siri” and then another beep beep after you finish your statement. My only advice is that you can prevent the first beep beep by going straight into your statement after saying Hey Siri, as one continuous statement. It will still capture everything you say.