EDIT: Okay, sorry, I did misunderstand what this was about. I never had any exposure to "scale harmonizations", the "rule of the octave", etc. when I was a student—I learned tonal harmony in terms of chord construction and voicing, voice leading, characteristic progressions, modulation techniques, and large-scale structure. We did use figured bass, but not in that fashion, and when we talked about harmonizing melodies it was in terms of the entire edifice of tonal harmony, not something you could distill down to a characteristic set of chords over a scale. I'm not sure that that "partimenti-style" teaching approach was all that common outside of the Neopolitan tradition in the 18th century (although if you have references to the contrary I'd be curious to see them). I'm having a hard time finding much rigorous information on it outside of Robert Gjerdigen's website, so that what I'm working from.
Notably, Quantz doesn't seem to have thought in these terms either, at least not as far as he lets on in that book. I flipped through Edward Reilly's translation and read the chapter the examples you cite are from, and he doesn't seem to ever present material the way those Neopolitan theorists do. Of course, he's not writing a composition manual—those are tips for beginning flutists to start on devising melodic variations when improvising and the like:
Since...most musicians lack the necessary instruction...so many incorrect and awkward ideas appear that it would be better in many cases to play the melody as the composer has set it rather than spoil it repeatedly with such wretched variations.
To remedy this abuse somewhat, I shall give some instructions, for those who still lack the requisite knowledge, as to how variations may be made in diverse ways on plain notes in the majority of the most common intervals, without contravening the harmony of the bass.
I do not pretend to have created in these few examples all the variations it is possible to discover on these intervals; I present them only as an introduction for the novice.
So, if we want to tease out a Neopolitan-style rule of the octave from his examples, we're going to have to make some pretty subjective decisions. He's not trying to present harmony in a systematic fashion like that, so we have to try to read his mind a bit; as I said in my original answer, those examples don't necessarily present progressions you could extend to a general case that way. What's more, I'm not sure he even thought about music in the same fashion exactly—again, maybe, but if so he doesn't let on. So we're kind of translating between one musical language and another, even if they do have a lot in common. I'll do my best to indicate what I think is more-or-less an objective truth and what's more a matter of opinion so that you can work based on your own tastes.
Anyway, the biggest problem with lifting directly from Quantz's examples is that he elides the rules of good voice leading. This is understandable because he's just sketching out harmonic frameworks for flutists to devise variations in—the flutists are meant to work out details like that themselves (see e.g. the other part of his Tab. X). However, rules of the octave are meant to provide guidance for writing harmony, so they do worry about voice leading, as you can easily verify from the manuals Gjerdigen put up. So, despite my earlier confusion, much of what I said is still relevant, although I'll do my best this time to put it in the proper context.
A few points in particular: Quantz doesn't concern himself much with avoiding parallel 5ths and 8ves, keeping the motion of individual voices varied, or balancing parallel and contrary motion between voices. We will need to correct for all of this somehow (there are some exceptions for parallel 5ths and 8ves as far as the Neopolitan folks go, which I'll discuss). None of that is just my opinion: on top of examining the rules of the octave given in the books Gjerdigen put up, you can read Fux's Gradus ad Parnassum for a text that was popular at the time and that states them quite clearly.
Also, rules of the octave are designed around harmonizing a descending or ascending bassline, not a melody. The partimenti exercises that come after they've been learned are designed around harmonizing a melody, but rules of the octave are meant to help students make connections between basslines and the harmonies implied by them. So, instead of working from Quantz's melodies, we'll need to work from the appropriate basslines, and see if we can harmonize them in accordance with Quantz's methods.
One last point: all the Neopolitan theorists seem to give three rules of the octave per "mood", starting with the 8ve, 3rd, and 5th on top of the I chord respectively. Since you only set out to write one, I'm going to do the same, in the interest of saving myself time. More-or-less arbitrarily I'll do one starting with the 8ve on top.
Just to make sure we don't stray too far from the Neopolitan tradition, let's do a quick analysis of one of their rules of the octave—I went with one of Fenaroli's:
I did this in a rush so if there are any careless mistakes I apologize. :P
From this (and the other Neopolitan rules of the octave you can read about on that website) we can glean the following:
- Always start and end with a I chord in root position.
- End with a V-I, but it's okay if it's quite weak in cadential terms—the stepwise bassline forces you to make some sacrifices (cadences are taught separately anyway in this tradition).
- You can handle some of the awkwardness of the descending bassline through a brief tonicization of the dominant key starting from the third step (other Neopolitan theorists do this as well).
- Occasional parallel 5ths/8ves are permitted between the bassline and the inner voices, but never sequentially. (Apparent parallel 5ths/8ves in the upper voices are generally due to the entrance of a 4th voice for the sake of 7th chords and such.)
- The inner voices can be more restricted in their movement than they might be in a real piece, but they should still have pleasing contours.
- The "melody" in the top voice does not need to follow the rules of good melodic writing; it can be thought of like another inner voice (all of this is meant as a study of accompaniment).
- EDIT 2: In my haste, I neglected an important point, which is that the chords assigned to step 5 of the scale should be appropriate for cadential purposes. Of course, when composing an actual piece of music, you could always substitute in the appropriate harmonies when you wanted a cadence, but these "rules of the octave" seem to be meant as a starting point for teaching purposes, so it's better to keep things simple. The Neopolitan theorists seem to always put a V chord there, and furthermore a triad as opposed to a seventh chord.
- Other than these guidelines, follow the rules of good voice leading and keep the harmony moving.
Okay, now let's check out Quantz:
- Tab. IX Fig. 2 steps 1-2-3 is basically just a long I chord (the apparent ii in the middle is basically passing motion on the way to a first inversion I). We don't really have the space for that here so we might substitute a vii as Funaroli uses, which shares some notes with the ii but will create a greater sense of motion.
- Tab. X Fig. 4 steps 3-4-5-6 is I-vii-I-ii7-V. This gives us some trouble, both because using a vii on steps 2 and 4 will get monotonous and because we should avoid returning to the I after step 3 for the same reason. The vii forces us to move to the I next for voice leading reasons, so we can solve both problems by substituting something else. V7-vi7-ii7-IV might be the closest we can get, since we need to follow this with a V-I.
- Tab. XI Fig. 6 is either steps 3-4-5 or 6-7-8 depending on whether you view this as in C or in G. In C terms it's vi7-II-V; in G terms it's ii7-V-I. On the ascent it doesn't really make sense to tonicize G, but we are planning a vi7-ii7-IV-V progression which is fairly close. On the descent we will tonicize G, but we can't really use this progression, both because we'll need to already be in G by step 5 (downwards) and because we'll have a B in the bassline on step 7.
- Tab. X Fig. 5 steps 6-5-4-3-2 is IV-I-vii-I-V. We would need to transpose this into G for the first two chords. Sadly we can't use a IV (in G) at this point because of the A in the bassline on step 6, and we kind of have to use a V (in G) there anyway in order to move to G. We'll also need to do something else for the vii-I-V part because we need to get back to G on step 2. We could do IV-I-V (in C terms) which isn't so far away.
Adding all this up, we get:
EDIT 2: I realize this has a significant issue if you were to actually use this to try to compose 18th-century-style music, which is that the chord given for the 5th scale step in the ascending rule is a vi7 chord, and thus would not perform the appropriate dominant function in perfect cadences and things. However, I'm reluctant to change it, because we live in the 21st century and I love the sound of that chord very much, so I would happily use it compose with anyway, broken tonal functionality or no. But I will revise this to be more suitable for 18th-century-style composition purposes if you like.