Hi Bryan,
I know very little about pickleball, so I am not the one to ask when it comes to the practicalities of running competitions, but I can help with the mathematical aspects. I have never heard of the Q program (can you provide a link?), but I am familiar with the math of "16,4", in fact what I had in mind when I mentioned the 25 players is very similar. Have a look at the
schedule here for 25 players labelled A to Y. This becomes your court allocation schedule, and each week in the schedule becomes a period of your competition. For example in period 2 you allocate the 5 players (C H M R W) to court 3 and there they play the mini doubles tournament:
(1 2) vs (3 5)
(2 3) vs (4 1)
(3 4) vs (5 2)
(4 5) vs (1 3)
(5 1) vs (2 4)
where 1=C, 2=H, 3=M, 4=R, 5=W. Just follow the same pattern for all 6 periods and 5 courts and you will have a schedule where all pairs of players partner once and oppose twice. If you can't manage 6 periods then you could cut it short and you would still have optimal mixing within the constraints of your send-5-to-a-court format. Hope that helps.
Ian