Can you get QMP for having a no go on character?

If your NO GO isn't warranted you can immediately appeal or rebuttal your NCOER. From what I've seen, character violations like integrity issues and DUIs don't get retained, and competence issues usually do. Letters of recommendation - from you chain of command, not your peers - carry a lot of weight with the board. Except when it's a DUI, if you get referred for a DUI just start working on your MEB process so you can hopefully MEB out instead since medical separation offers higher separation pay.

The Character section of Part IV on the NCOER encompasses what the Army Values portion did before, plus more, so I imagine it can still possibly influence a QMP review at or shortly after the next promotion board. As to how easy it is to fight it, that would be better addressed by someone who is still currently in. My go to for all answers is SFC (Join to see) .

First question I have is “do your quarterly counseling’s support this rating?”

If you were counseled on this and made no improvements, fighting the NCOER may be difficult. If they slammed you on your NCOER without supporting counselings you probably have a strong case to take up the chain.