It's not really very plausible, no. HBO is, not surprisingly, suffering from a severe lack of understanding of ancient religion in general and definitely in Roman religion. Which is understandable--ancient religion is one of the most complicated and difficult to understand aspects of classical studies.

The big thing that stands out for me is that HBO doesn't seem to understand the difference between Greek and Roman religion. We're taught in school to think of them as basically the same thing with different names. That's not really true at all, even on the very superficial level that we introduce students to. It's just convenient. Roman religion is quite distinct from all forms of Greek worship and follows some very different, and often quite bizarre, paths, frequently centering around rituals which a Greek would (and usually did) find ridiculous and difficult to comprehend. The most obvious thing here is that HBO has Augustus talk about anthropomorphic gods interfering in human affairs. While the Greeks, from Homer on, had a well-developed sense of anthropomorphism, Roman religion never really fully developed anthropomorphic gods. In earliest times during the Archaic Period Roman religious worship centered on forces of nature that were ill-defined with poorly laid-out powers and demarcations. It seems that the Etruscan influence on Rome was the first real influence in the realm of anthropomorphic gods with well-defined characteristics and powers. The Etruscan influence was responsible for the introduction of the cult of Jupiter Optimus Maximus and the Capitoline Triad of Capitoline Jove, Capitoline Juno, and Capitoline Minerva. The Capitoline Triad (Jupiter Optimus Maximus is the same as Capitoline Jove) was fully anthropomorphic, although it's debated just how firmly nailed-down their powers were (they seem to have been three aspects of the same divine force protecting the city and its people, rather than three separate gods with powers separate and distinct from each other). However, even though this was one of the oldest cults in Rome it had only marginal influence in other cults. Most other cults of Jupiter, for example, had no concept of anthropomorphism. For example, the cult of Jupiter Victor or the cult of Jupiter Stator were not involved in the worship of some anthropomorphised figure, but in the power that the god described. And the existence of the earliest Capitoline Triad, predating the Etruscans, of Jove, Mars, and Quirinus is quite telling. Neither Quirinus nor Mars were ever really anthropomorphised. Mars was sort of anthropomorphised, but not quite, since he was really quite different from Greek Ares, and Quirinus was a protector-force. Its usually thought that Mars and Quirinus were the protector deities of two separate groups living in the city at that time, although other explanations (most of which are not mutually exclusive) also bounce around.

Now this would be much more plausible if Octavian had been a Greek. Not just the concept of anthropomorphic gods, but all of it. The statement you've paraphrased here is quite clearly influenced by the philosophical principles of the late Classical Period and the Hellenistic Period in Greece, particularly by the Cynics, Stoics, and Epicureans. Now these are all schools with which any educated Roman would be familiar with, but with the exception of the Stoics and to some extent the Epicureans they didn't have that much influence on most Romans during the 1st Century, B.C. The idea of the "prime mover" sounds suspiciously like HBO is referring to the concept of the Stoic God, which encompasses all and is everything, but that's not the way the Stoic God worked. The Stoic God was a sexless, formless being of all good and evil that didn't interfere with the world and was really only present specifically through people's minds. Stoicism had a profound influence on many Romans of the upper class, since it taught that a person's inner bein, i.e. the workings of his mind, were more important than anything else. Seneca, probably the most well-known (and the most obnoxious) of the Stoics, is constantly going on about how the activities of the crowd and the activities of a good man's mind have nothing to do with each other and how the good man is completely self-sufficient. But although the Stoics were very influential, such an impious attitude is pretty antithetical to their teachings. The Stoics believed that maintenance of tradition, piety, and responsibility were all attributes of a happy, self-sufficient person, and they believed that the other gods existed, interfered in people's lives, and were (or could be) anthropomorphic, just that they were all attributes of the Stoic God. The Epicureans were also of great influence in some circles. Catullus, famously, was an Epicurean. The Epicureans believed that everything was composed of atoms and that supernatural forces therefore didn't exist, although really weird stuff could happen if atoms were arranged in a certain way. They believed that the gods very much existed, but they didn't really believe in divine power as being some sort of supernatural force outside the realm of all universal understanding. Instead the gods were composed of atoms, and their power was a result of atomic arrangements, so theoretically at least they weren't much different than humans. The big difference between gods and humans is that while the forces that hold human souls in their bodies cannot keep their souls in forever (and thus we die and our souls break apart), the souls of gods were inalienable parts of their bodies and thus they were both immortal and blessed. Opinions on Epicureanism in modern times is confusing, mostly because the early Christians completely warped the Epicurean school, conflating it with hedonistic heresies. Because Epicureanism, too, believed that although the gods don't really interfere with human affairs, they have material bodies (which the Epicureans usually assumed looked human) and were immortal and blessed and should therefore be respected for being immortal and blessed. But all of these philosophical principles as adopted among the Roman nobility were very unlike their Greek counterparts in many ways. They were fitting into a society that was very different and operated along different ideas of morality.

So what about Augustus the man? Augustus made a big show in his later life of being the defender of old Roman morality, much of which had been conflated during the period after the Punic Wars with Stoicism and which was highly influenced by Stoic teachings. But just how far did Augustus actually believe all that? Certainly his policy was colored by his desire to set his image as that of the "Benevolent Dicator" who put an end to civil war and restored Roman morality that Victorian scholars labelled him as being, rather than the tyrannical, power-hungry destroyer of liberty that Syme compared to Hitler. Whether he really believed any of this by the end of his life is difficult to say. Octavian's youth had been spent breaking pretty much all of the rules (except maybe for filial piety, but that was really a means to an end) that he tried to hold people and himself to so strictly in later life. And Augustus only set himself up as being some sort of moral paragon well after he had firmly gripped the entire Roman state in his grasp. We have no evidence for what Augustus believed privately, only his public statements and the evidence of observers, whose testimony is difficult to really analyze because many of them were either in Augustus' employ or were worried about pissing him off. What we do have is other people's opinion of him. Much of the great literature of the Augustan Period is overtly pro-Augustan, but runs very contrary to Augustus' preferred image if you examine it closely enough. Virgil, for example, who on the surface is one of Augustus' great supporters, quite obviously paints Augustus to be a villain at many points, and frequently suggests that he is immoral and ruthless, willing to do anything to get and keep his power.

So the short version is that it's complicated. We have no way of knowing what Augustus personally believed and anything we can say is really conjecture. It's highly unlikely that Octavian would ever say something quite like what HBO has him saying, simply because such ideas didn't really exist yet. When they did exist they didn't take that form. Not to mention the fact that they have Octavian making a statement that shows the influence of two schools that contradict each other, and that the Romans and Greeks alike saw as largely antithetical.