No. (Updated)
This boils down to the following tests. Could the Romans:
- Keep a large octopus alive in captivity for long periods of time?
- Contain and transport live pelagic creatures aboard a ship of the era?
- Ever sail a ship from Rome to the Red Sea and back?
- Conduct trade with East Africa?
Previous answers have treated these questions like a checklist: if we can say "yes" to all of them, the answer to the top-level question must also be yes. Yet these aren't separate questions: when combined, the difficulty increases exponentially, to the point where I'm confident that the Romans could not achieve all of them together.
Keeping the octopus alive: they're not just fish with arms
Speaking from an aquarist's perspective, keeping octopuses (and cuttlefish) poses some distinct challenges compared to fish and other aquatic life. They're quite delicate creatures and can easily be killed in transport.
Keeping the water cold
GPOs prefer 10C water and will tolerate up to 12-13C. Your octopuses seem to be hardier in this regard, but how high can they go?
Keeping the water fresh
See e.g. Keeping cephalopods in captivity:
An octopus produces approximately three times more ammonia than a fish of a similar mass; partly due to its having three hearts and therefore three times the oxygen requirement of fish. As a result oxygen levels should always be kept as high as possible. To do this (and to remove all the excess waste) always use oversized filters and skimmers.
The most significant challenge: when they're stressed or perceive danger, they eject a cloud of ink. In an aquarium, this can quickly suffocate them unless the water is immediately scrubbed:
When an octopus feels threatened it may eject viscous ink as a smoke screen. Although it is not poisonous it can and will coat gills and this may lead to asphyxiation. The ink can often be removed by catching it in a fine net but protein skimmers and good quality carbon is a must have! Always work slowly to avoid startling a new octopus.
Perhaps your sapient octopuses can suppress this reflex, but for how long?
Feeding the octopus
They also need live food on a regular basis. GPOs can survive for long periods of time with no food, but this will increase their stress level.
Perhaps your sapient octopuses are willing to take what they can get.
Containing the octopus
A GPO needs a container of sufficient size in order to survive. The 50 gal pithoi mentioned in comments will only suffice for short periods of time; for long water journeys, you're looking at effectively building a long-term aquarium onboard, with all the attendant difficulties.
A GPO can escape through a two-inch hole: its only hard body part is its beak. They're fantastic at escaping, and human-level intelligence would only make them more so.
Heavy metals in the water
Given the metals the Romans had access to, they'd have immense difficulty building a containment vessel and associated hardware for the octopus that wouldn't poison it, considering that even a trace level of many common metals is fatal to them. There are examples in the aquarium community of cephalopods killed when their keeper put their hands in the water after touching a copper object, for example. Modern tap water that far exceeds EPA standards will readily kill invertebrates if the copper in it is not chemically detoxified. Tin is also toxic.
(Lead is also hazardous, though "what lead concentration is safe for a captive octopus" is understandably difficult to research in the modern literature.)
Considering that bronze played such a prominent role in Roman tools, and that Roman pumps and pipes were fashioned out of bronze and lead, it's hard to imagine they'd adhere to safe handling standards in this regard.
Putting this all together
Assuming this was attempted, my guess would be that the octopus would sicken and die within days, in a container that could transport live fish indefinitely with no problems. This is the modal experience for naive or unprepared octopus-keepers, even with modern aquarium technology, after all.
Even with full access to modern technology and marine science, there are many species that we cannot keep alive in aquariums. Imagine this scenario: the Romans capture a sapient octopus that doesn't want to be caught, put it in a tank on board the ship, somehow know not to use any of their most common metals in any pumps or fixtures or pipes or fasteners, are capable of pumping enough water at the right temperature and salinity through the tank to keep it alive... and then it dies because someone used a copper mug to drop its food into the tank several days ago. What's more likely: that the Romans, further from home than almost any of their compatriots have ever traveled, in barely-known territory aboard a round-hulled trading ship, would sit down and conduct a thorough post-incident review and determine the chemical cause of death, and then go and catch another octopus? Or would they conclude "octopuses just die when you put them in a tank and we don't know why"?
There's a lot more complexity I'm not getting into here. For further reading, see:
Keeping aquatic animals alive on board ship
While the Romans did have shipborne tanks for transporting live fish (naves vivariae), the Grado wreck is the only extant example of what may be a shipborne pump for refreshing the water, and that is disputed. See Comment on a Recent Article Concerning the Hydraulic System of the Roman Wreck at Grado, Gorizia, Italy, cited from the Maritime hydraulics in antiquity Wikipedia article. As the article states, no other naves vivariae had this capability, and it's quite likely the pump was used for something else.
Seawater to freshwater
Nonetheless, this relies on the immediate and constant availability of native seawater that sustains the animal. Even granting the existence, availability, and function of the Canal of the Pharaohs, the hypothesized route via this canal does not cross from the Red Sea directly to the Mediterranean (the Suez route). Instead, it traverses the Nile, which is freshwater. The rapid desalination of a saltwater habitat will kill almost any marine animals inside; even a minor fluctuation of salinity will do it. Octopus in particular are vulnerable. I found this article when researching lead levels in octopus: "
Abnormal mortality of octopus after a storm water event":
In summary, the results in this study showed that a short-term runoff event might change abruptly the salinity leading to the disruption of the osmoregulation function of octopus and consequently leading to its death.
In any case, "don't put freshwater in your saltwater tank unless you want everything in there to die" is Aquariums 101.
Can the Romans transport enough seawater to keep it alive for the weeks it takes to cross? Will recirculating that seawater do it? Not unless they've invented filtration of ammonia and other toxins. Can they salinate seawater to the right levels, with no test equipment (nontrivial even with modern tech, to the point where people buy seawater by the gallon for their reef tanks)? I'm maximally skeptical.
Getting there at all
The cited document, the Periplus of the Erythraean Sea, does indeed describe Roman contact and trade with East Africa. I'm not a classicist, but if you dig further into the Wikipedia article, it mentions that the author likely originated from the Roman port of Berenice; that is, on the eastern coast of Egypt itself.
Just because
- the Romans traded via ship,
- the Romans traded goods with East-Central Africa and the Indian subcontinent,
- the Romans ever successfully took ships from the Mediterranean to the Red Sea,
it doesn't follow that the conjunction must be true, that unbroken Roman maritime trade occurred between East-Central Africa and Egypt. The red dotted line on the Wikipedia Periplus map is misleading: goods would change hands, and transports, many times over the course of the journey. See this map, from the article on Roman commerce:
Reading further into the Indo-Roman trade relations article makes it clear that the primary Roman trading ports were along the eastern coast of Egypt. My research has found no evidence of any maritime trade between the Mediterranean and the Red Sea/Indian Ocean. This article states:
Roman trade through the Red Sea corridor began in Ostia or Puteoli from which ships traveled to Alexandria and then Coptos on the Nile. It took about 20 days for the ships to reach Alexandria from Italy and another 11-12 days to move the goods down the Nile to Coptos[...]
The merchandise arriving at Coptos was carried overland by camel caravans to the Red Sea ports of Berenike and Myos Hormos. It took about seven days for the caravans to make their way to Myos Hormos (177 km or 110 mi) and twelve days to more southerly Berenike (370 km or 230 mi).[...]
Final note
GPOs really aren't that big and wouldn't be particularly spectacular to fight. They may look big in some photos, especially with their tentacles and web spread, but they're realistically a basketball-sized body connected to some tentacles considerably smaller than your arm. See this absolutely adorable video of the octopus aquarist at the Monterey Bay Aquarium caring for Chica the GPO.