I am not aware of any tools similar to Ian's that have been written for QGIS (in fact all the tools I know of including commercial are specifically for ArcGIS). His script is written in Python though it does use arcpy, and his source and process description are pretty thorough. You might contact him via his blog (or he may chime in, he's on here) and he might be willing to help adapt it.
The method used may vary depending on what data you start with. If you only have sections, you'll need to go 16ths. If you have quarters, you only need to go quarters. Official data usually doesn't contain quarters though, it goes straight from section (aka First Division) to quarter-quarters (aka Second Division). And further it depends on the data itself - Ian's method utilizes vertices in the polygon to start with. If you don't have those, you have to look at midpoint/quartering each line to make the grid. Or possibly making a regular grid and running something like ArcGIS's Snap tool to fit/adjust it, otherwise a regular grid tool such as those you link to cannot be used.
You end up with an approximation in any case, but from your question it sounds like you're just going for 'close enough'.
If you already have an existing grid and need to preserve it to ensure it lines up with other data created based on it, then there's no way around doing your own quartering. However if you can, downloading an 'official' grid can save considerable headache (note of course that not even 'official' grids will always match because different entities do different updates, adjustments, and corrections). There are different places to find PLSS data sets. You've mentioned Montana specifically. Apparently the State Library took over for the BLM as the PLSS data custodians last spring (2014). You can now find the most current, up-to-date version of the grid at:
May be the same as the data linked in your question, but I specifically went to the 'recommended for additional information' link. I do note that this dataset does provide Second Division level, but it's only a 'partial' file at that level. I believe it does or should have everything down to section level based on a quick pan of the data, and the quarter-quarter level is pretty extensive - I'd estimate at least 80%. In part this is because the PLSS isn't actually finished in parts of MT - there are areas that have never actually been surveyed.
You probably already know, but just for the benefit of others (and to clean up the mess of comments), the PLSS grid isn't regular. Furthermore not all quarters actually divide into quarter-quarters - in some cases there are Lots interspersed. Basically the PLSS is nominally a grid, but the further you get into the details the messier it gets. This pdf document has extensive information on the parts of the grid and some graphic examples - there's a lot of good info in it.
