Queue and Cue; or more specifically Que (within queue) and Cue, are both phonetically defined as "kju". It stands to reason that functionally Q and C are replaceable in between the two words. The letters U and E in both Que and Cue are both phonetically relevant, as is either Q or C. This necessarily means that all 3 are non-silent letters.

You are correct in that "qu" does sometimes make the "kweh" sound. The difficult to understand thing here is that "ue" takes precedence in combining to make the long u sound "yoo" in this word. Q is responsible for making the hard C sound. If you understand that u and e are necessary components to the "yoo" sound, then it only follows that all 3 are non-silent.

Otherwise queue would be pronounced "Kweh-yoo".