I'm not an archaeologist or architectual historian, but I know and have worked with a few of the textual sources from the period. I've just now briefly skimmed through some of the more immediately available sources, and the short answer is probably yes, though perhaps not as early as the sixth and seventh centuries. What is known as the apokrisiarieion (ἀποκρισιαρείον), has been translated both as "the Envoy's quarters" and as "the Bureau of the Emissaries". As far as I can see it is first mentioned in the Kletorologion by Philothetos (899), a list of various offices of the imperial court that mentions the Warden of the Apokrisiarieion (Bury's 1911 edition with commentary here, check p. 93). Most scholars seem to accept that it is a building specifically for lodging foreign envoys, but it might also have been a bureau responsible for tending to foreign envoys. Interestingly, I havent been able to find a primary source describing housing anyone in the apokrisiarieion, although absence of evidence is not evidence of absence.

In the article 'The Logothete of the Drome in the Middle Byzantine Period' D. A. Miller seem to accept that it is a building for foreign envoys writing this about it: "The location of the Apokrisiarieion is not definitely known ; Liutprand says that it was unconscionably far from the Great Palace, and that he was able to see the fleet sail from his quarters. Perhaps it was located near where the Italian factories would be located later : on the Golden Horn in the southeastern part of the City." (article in Byzantion, 1966, p. 447).

Even though we don't know exactly where it was we know something about how ambassadors were treated and how the lodgings were experienced. Liutprand of Cremona, mentioned by Miller, was an envoy of the German Emperor Otto II, and visited Constantinople in 968. He mentions practice of keeping ambassadors under close guard, and, perhaps for reasons of security, they were intentionally kept in lodgings far away from the imperial palace:

"We were shut up in a palace large enough, indeed, but uncovered, neither keeping out the cold nor warding off the heat. Armed soldiers were made to stand guard who were to prevent all of my companions from going out and all others from coming in. This dwelling, into which we alone who were shut up could pass, was so far removed from the palace that our breath was taken away when we walked there - we did not ride." (From E. F. Henderson, Select Historical Documents of the Middle Ages, (London: George Bell, 1910), pp. 440-477).

Lodgings far away, then, and not particularly comfortable to a German bishop and ambassador. Liutprand is complaining about this in a letter to Emperor Otto II. But perhaps Liutprand exaggerated the conditions in Constantinople to exalt his own emperor. The western emperors had a tenous claim to the title, and if Constantinople and its emperor were made to seem worse than they actually were, perhaps it would flatter Otto and legitimise his claim to the title? An Abassid ambassador that visited Constantinople in the late tenth century had no complaints about the lodgings:

"So I proceeded to Constantinople and made my entry after I had been met and most courteously escorted by court officials. I was honourably lodged in the palace of the Kanikleios Nicephorus (the envoy come with me) who stood in favour with the Sovereign. " (From H. Amedroz, "An Embassy front Bagdad to the Emperor Basil II," Journal of the Royal Asiatic Society, (1914), pp. 921 25. Reprinted in Deno Geanokoplos, Byzantium, (Chicago: 1984), 339-340)

You might notice, though, that the Abassid ambassador was not lodged in the apokrisiarieion, buth rather in the private palace of the Kanikleios Nicephorus. The Kanikleios was a senior court officer and keeper of the imperial inkstand. In fact, it seems like the upkeep of ambassadors were very much an ad hoc procedure, depending on the importance of the person in question. The book of ceremonies (De Ceremoniis), probably written in the 950s, but retaining chapters from Peter the Patrician (c. 500-565) confirms this in a description of the procedure of receiving a senior Persian ambassador:

"His lodging in the City should be prepared in advance according to the rank of the person and the size of the retinue he is bringing, and there should be ready there beds and bedding and ovens and braziers and tables and workers to fetch water and to assist with the other menial household services." (From Constantine Porphyrogennetos: The Book of Ceremonies, translated by Ann Moffatt and Maxeme Hall (Leiden: 2017) p. 401)

The sixth-century source, preserved in a tenth-century book of ceremonies, then does not specify that foreign envoys be kept in any specific building, only that it should be appropriate to the rank and size of the retinue. Similarly The Book of Ceremonies does not specify a specific lodging if ambassadors arrives from an emperor proclaimed in the west:

"When ambassadors are coming, the magistros should learn in advance and get ready their lodgings and also send someone to meet them and conduct each to bis lodgings. They also prepare bedding in advance and money for their expenses. They settle into their lodgings, and after one or two days they see the magistros." (From The Book of Ceremonies, (2017) p. 393).

It seems clear though that ambassadors were normally kept under close guard, not only in the tenth century with Liutprand of Cremona, but also in the sixth century. In the work on the Persian wars, the author Procopius (c. 500- c. 565) does indicate something about the regular practices in the recption of ambassadors, in his description of the irregular reception of the envoy Isdigousnas, sent by the Persian Shahanshah Khosrow (r. 531-579):

"And this man, unlike all other ambassadors, did not have the experience of being under guard in any sense, but both he himself and all who followed him — and they were an exceedingly numerous company — enjoyed complete freedom for a long period of time in meeting and associating with whom they wished, walking about in every part of the city, buying and selling whatever they pleased, and carrying on all manner of transactions and devoting themselves with complete unconcern to the business connected therewith, just as they would in a city of their own, with not a Roman following or accompanying them at all or deigning to watch them, as is customary." (From The Wars of Procopius 8.15.20, (Cambridge, MA: Loeb Classical Library, 1928), pp. 214-16).

To summarise, by the late ninth century, and probably before there was an institution or house known as apokrisiarieion, which probably housed foreign envoys, but I have not been able to find sources stating that any envoy actually stayed there. If Liutprand of Cremona was one of its lodgers in 968, it most likely far away from the Great Palace. We also have sources indicating that ambassadors stayed somewhere else than in the apokrisiarieion, such as the Abassid ambassador arriving in the 900s, and staying with the Kanikleios in his palace. Finally, neither texts from the sixth century nor texts from the tenth century prescribe any specific house for ambassadors. The sixth century source Peter the Patriarch instead states a principle that ambassadors are lodge according to their rank, and most likely their according to their importance to the Byzantine government. That the Byzantines provided special service to certain envoys can be seen in the case of Isdigousnas. Procopius relates how the normal practice of keeping ambassadors under guard was waived in his case, and perhaps it would not be unreasonable to assume that he was provided with especially fancy lodgings too, even though Procopius says nothing about the matter in any of his works.