Q-Scale is 1/4 inch scale utilizing a track gauge of 1 3/16 inches. Q-Scale is close to Proto 48 but not quite there.
https://oscalekings.org/WP/2-rail-o-scale-primer/
"Some other modelers used a track gauge of 1-3/16″, calling it Q gauge, which was close to the actual dimension. One famous Q-gauge modeler was Minton Cronkhite who built the original O scale layout for the Chicago Museum of Science and Industry."
17/64 inch scale has historically been O-gauge/O-scale and utilizes O-gauge 1 1/4 inch track. At 17/64 inch scale O-gauge track scales out to be 4 feet 8 and 1/2 inches which is standard gauge in the prototype world. At 1/4 inch scale, O-gauge track is a scale 5-feet in gauge. Good for Russia and some other countries, but not for North America and Europe which use standard gauge.
"........we should point out that the 1-1/4″ track gauge commonly used in O scale actually measures out to 5′ rather than the correct 4′ 8-1/2″. This came about because early toy train manufacturers used the 1-1/4″ track gauge (which was invented by Marklin in the 1890s) since it was a nice round number, and the small gauge discrepancy was not very noticeable. Some early O scale modelers used 17/64″ to one foot as an exact scale for the 1-1/4″ track gauge........."
When you look at old publications and ads this is how it is approached.
Now to today. Somehow 17/64" inch scale and "Q" scale has become intertwined. I am not sure how that happened but the labeling of 17/64" inch O-scale as "Q-scale" is a recent event.
https://dfarq.homeip.net/q-scale-model-railroading/
On a side note, Lionel's first O-scale strains such as the M10000 were scaled to 17/64.
Back in the day of slide rules and such, 17/64 was not an easy fraction to work with, but today's computers handle 17/64 just as easily as 1/4.