I am a German national. My background is in engineering and business, but I am interested in history. I am convinced you can find literature on both the traditional North-South difference, as well as the more recent East-West division of the country.
For the North-South difference, you have to go back to the time of the reformation of the Church (2017 is celebrated as the Luther year, 500 years ago).
For the East-West difference you only have to go back to 1990, or not even, because the effects of the re-unification of Germany have not fully been overcome until this day, and can still be felt as I am writing this.
That is maybe also the reason why this element is more present in the public sentiment:
With all these arguments you can see why today's Germans are more occupied by this than their traditional North-South (and therefore mainly religious) divide.
Germans in general are Christians (including Austrians). But the more you go north, the more protestants are there (after Martin Luther's reformation, exactly 500 years ago in 2017). The South is catholic, eg Bavaria. The same is true for Austria .
In fact, the bloodiest war ever fought on German soil was not WW 1 (outside Germany) or WW 2. It was the 30-years war from 1618 - 1648. No other war cost so many lives / population. Entire provinces were deserted, animals like wolves came back ....
Light brown: areas with at least 1/3 of population dead
Dark brown: at least 2/3 of population dead.
As so often, many reasons and power struggles were mixed, but it was fought strictly following religious lines, Catholics against Protestants. The main Catholic leaders were the the Austrian Emperor of the house of Habsburg, the arch duke of Bavaria, and of course generals (Wallenstein and Tilly respectively).
On the Protestant side, there was no such clear leadership at first within Germany, so at first, Denmark's king got involved, later the Swedish king Gustav II Adolf became the dominant leader of the Northern Protestants.
On this map you can see that "Germany" was really a big mess of small and smallest states.
Austria is the big yellow thing in the bottom right corner. Prussia is the tiny blue island inside the red Poland-Lithuania in the upper right hand corner. Of no significance at the time!
When this was over, after 30 years of bloodshed, without a winner, only losers, it was clear that the religious division would remain (much had been about freedom to exercise the protestant faith).
The patchwork Germany continued to Bismarcks age. In the meantime, Prussia grew into a major European power, to rival England, France, Spain Austria and Russia. But it was always much less than all of. Nonetheless, it's king Friderich II (called the Great for his talent as general) fought aggressive wars against Austria, for pure lust of conquest.
The religious divide continued to be so strong, that the French Emperor Napoleon III (Himself a catholic like all French) thought still in 1870, when he attacked Prussia (and the northern German states), that the Southern, Catholic, German manner would actually side with France, not with Germany / Prussia, and this even after the liberation wars to kick out the French of Napoleon I (Bonaparte) out of Germany which were still in the living memory of the people back than.
(I am not aware of anything similar being anticipated, or even happening in WW I, as the question suggests)
For Bismarck, holding together Germany (even without Austria) was more realistic to achieve than to have the eternal internal power struggle with Austria. Bavaria, the next largest catholic state, was already enough to handle. Bismarck wanted and forced the war with Austria to mark the dividing line.
As you can see here, the basic distribution is still valid today (people have not moved that much).
Protestants have the majority in the North, Catholics in the South.
But what is very visible too: just 40 years of atheist communism have eliminated either Christian faith quite substantially in the East German states: Athesism (no religious belief) has the majority there.
It must be noted that these numbers will be based on official Church membership. This membership costs taxes, and makes it more unlikely for East Germans to re-enter the Church, even if they personally have a belief (just as much as West Germans might still be in the Church without actually going to the service).
Having said all this, I think it is important to note that the rivalries today are all minor. While until 1870, states might go to war with each other, today's rivalries are much more like those of neighboring football clubs: People in Cologne will most emphatically point out they are not to be confused with Düsseldorf, which is just 1h away, and vice versa.
But this is by no means a "terrible territorial conflict" as the question suggests.
Bavaria is most proud and most outspoken to be somewhat distinct. In return, most other Germans consider Bavarians as somewhat strange and smile upon them.
On the other hand: I lived 10 years in Munich, the capital of Bavaria (I am born in Hamburg). Today, there are predominantly people like me in Munich, not true Bavarians.
Both North-South and East-West disputes today are almost exclusively economic debates: Richer states pay for poorer states, and the richer ones naturally don't like that.
Religion, which was responsible or at the center of many conflicts, plays hardly any role today.
Mobility of the people takes away most of the remaining differences.