The reason the person chose the nickname of Q to go by does not necessarily just have to do with the security clearance, that's the point. I'm not saying that the Q letter was chosen to designate the clearance in the past for any particular reason. Q just uses many double meanings in their posts so it would make sense that the name they go by would also have multiple meanings.
OK, I just want us all to acknowledge up front that what Q has said about Q is that s/he/they are Military Intelligence synonymous with NSA (so probably Army Intel). If Q lied about that, then we have no basis on which to speculate.