Desert Rommel - Tumblr Posts
….. I don’t think so?
Just off the top of my head I can’t think of any Particular examples of Zeon as a whole oppressing their own people (IE other members of Zeon, which I’ll get to in a minute).
EDIT: I can’t think of any examples of Zeon *overtly* oppressing their own citizens, other than propaganda.

I would like to call out Desert Rommel from mobile suit Gundam ZZ. He camps out in a small desert town and is implied to be oppressing the people there. Rommel is in charge of a remnant group comprised of some Desert Zaku’s and Dwadges. He smacks one of his own men for caring about his family instead of giving his all in his attempts to revive Zeon, and is implied to be ruling the town with an iron fist, conducting raids on the outer settlements and Federation bases. Thing is, he’s not affiliated with Zeon at this point, since he’s active during ZZ, and his actions seem largely to illustrate just how desperate his remnant group is.

The Manga “The Plot to Assassinate Gihren” does have some depictions of Zeon’s secret police and has examples of how the population was kept in line (I shan’t spoil, not least because I haven’t read all of it). It does however probably fall under the header of “Zabi Plots”.
Generally, when it comes to their own population during the one year war, my understanding is that Zeon didn’t really have to oppress their own people as such, since they had an excellent propaganda ministry and the Zabi family (Gihren) were massively charismatic and didn’t really require much else to keep the general population in line. It’s implied (at least in the Origin) that Kycillia had a lot of pull with the “civilian” police, and presumably she would deal with any major dissenters. It is stated in some side materials that the Zabi’s suppressed certain political dissidents who were vocally opposed to their rule, but it’s never shown.

I would however note that the Iffish Colony Gassing (the Opening Stage of Operation British) could be taken as an absolutely massive case of this, since I understand that publicly Zeon was trying to do the whole “Independence from the Corrupt and Unfair Federation!” thing, which Iffish completely undercuts. Zeon slaughters billions of colony residents, it’s supposed allies, in order to use Island Iffish as a weapon against the Earth. If the Federation hadn’t fought the (very costly) Battle of Loum, they probably would have tried again, doubling the death toll.
Like, in regards to your average Zeon Colony resident, I expect they’d just be seeing an awful lot of Gihren’s speeches and the occasional riot suppression. Maybe some shady secret service guys.
Genuine question that has just occurred to me: to what extent, in Universal Century Gundam stories, are the various iterations of Zeon shown to oppress their own people?
Because that strikes me as an interesting wrinkle to the whole ‘are they really the bad guys’ line of thought. I know the Earth Federation is repeatedly shown to be oppressive towards its citizens (that is how we got into this mess in the first place, after all). And that’s aside from the unethical experimentation and so forth – they are shown to trample on the rights of anyone inconvenient, up to and including instituting a police-state (the Titans, the situation by the time of Gundam:Hathaway).
Whereas on the Zeon side, yes we have the multiple war crimes and mass destruction against the opposing side, but the only solid incident I can think of where Zeon’s own citizens were definitely chafing under their rule is the Cicero uprising in ZZ. Which is … interesting given the issues of legitimacy inherent to Harman’s plot-line.
Like the Federation, I’m sort of parceling off the cyber-Newtype malarkey in its own box, and I’d equally treat the ascent of the Zabis as a separate issue: this is more a question around how the regime they instituted is presented at the level of the ordinary person. I’m curious as to what the overall impression is among people more invested in the UC stuff than me?