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In Australian and New-Zealand English, the noun block (frequently with capital initial), preceded by the definite article the, has been used to designate a street or area in a city or town in which it is fashionable to promenade. This noun has frequently occurred in the phrase to do the block, meaning: to promenade in such a street or area.

The original reference was to a section of Collins Street in Melbourne (cf. below, quotations 1 & 2). The following explanations are from The Canberra Times (Canberra, Australian Capital Territory, Australia) of Wednesday 1^st May 1991 [page 20, column 1]:

Promenading coming back into fashion

MELBOURNE: Melburnians have been invited to step back 100 years, when “promenading” with sweethearts was more important than the Saturday footy game.
Melbourne City Council unveiled the new name yesterday for a pedestrian precinct in the heart of the city—Swanston Walk, formerly known as Swanston Street.
[…]
The Mayor, Richard Wynne, said the name “Swanston Walk” called on people to participate.
“That is what the whole project is about—people. For the first time people, not cars, will be the focus of Melbourne’s planning,” he said.
Council research showed that promenading down Swanston Street had been a favourite pursuit in the late 1800s and early 1900s, when the area was part of “The Block” and “doing the Block” was very fashionable.
The precinct was home to smart coffee houses, book-sellers and music shops, the ideal place for young people to meet.
It was de rigeur [sic] to be seen on Thursday and Saturday afternoons and in the 1870s Saturday football games were timed to start after Melburnians had finished lunching and promenading.

These are, in chronological order the earliest occurrences that I have found of this use of the noun block and of the phrase to do the block:

1-: From the column The Peripatetic Philosopher, by ‘Q.’, published in The Australasian (Melbourne, Victoria, Australia) of Saturday 6^th June 1868 [page 721, column 4]:

Truly the vanity of human nature is extraordinary. The other day I was slinking down the sunny side of Collins-street to avoid a bailiff whom I had espied admiring the fine arts at Wilkie and Webster’s window, and I was surprised to see so many people “doing the block.” The silks and velvets of the Victorian ladies gleamed resplendent. I saw young Fanfreluche airing the latest spolia opima torn from the bleeding body of his unhappy tailor. I saw numerous incarnations of insolvency parading in all the brightness of their borrowed plumes, and I wondered exceedingly. I wondered where the money comes from, with the crisis continuing, and squatting but the baseless fabric of a dream. I saw the well-built carriages and the ill-matched horses standing by the kerb-stone. I saw the usual afternoon’s Vanity Fair held, and I involuntarily attempted to estimate how long it will take Victoria to file a general schedule and begin again with a clean bill of health.

2-: From the column Topics of the Week, published in The Australasian (Melbourne, Victoria, Australia) of Saturday 13^th June 1868 [page 753, column 4]:

If you are a cynic and desire food for cynical rumination, you have but to “do the block” regularly, and you will find plenty of this kind of pabulum to your hand. Being a cynic, and having lived probably in a tub, you may not know what the “block” is. But it is a notable institution in Melbourne for all that. Some people appear to derive great pleasure in doing the block, and would no more miss it every afternoon than they would miss their dinner. But the spectacle you witness there is the reverse of cheerful, if only you look at it through a proper medium. It is easy to sneer at cynics, and speak of biliary overflows, jaundiced eyes, dyspeptic habits, sluggish livers, and hypochondriacal fancies, and, having done so, to think you have effectually disposed of those persons whose, no doubt, disagreeable habit prompts them to look below the surface. But the rottenness below exists, for all that it is suspected only by sallow-faced cynics. For is it not a truth that many of those carefully and expensively dressed persons who perambulate Collins-street, between Swanston-street and Queen-street, every day during the hours of from three to six, are insolvent either in pocket or reputation, and not a few in both? There is nothing immoral, to be sure, in promenading along a section of a street for two or three, or half-a-dozen hours, provided those who do so have no worse motive than bidding each other good-day, and procuring for themselves a certain amount of bodily exercise. Nor, for the matter of that, is there much to be objected to in people going to that quarter for the purpose of procuring such articles as are necessary from the tradesmen who have their businesses there. For shopkeepers must live, and men and women must be clothed and fed. But these are only secondary purposes in the minds of those who do the block. For the women are obviously bent upon maintaining a ruinous rivalry in dress, equipage, and generally expensive surroundings, and the men are content to be transformed into mere fringes upon the trains of the women. You are thus driven to reflect upon all the terrible waste of time and degradation of intellect involved in an occupation of this kind. If you could, like Asmodeus, take off the roofs of the houses of some of those resplendent creatures who float so gracefully and lightly over the pavement of the block, you would think very badly of the resplendent creatures, and you would forget the airy grace with which at first they enchanted you. For yon may have some old-fashioned notion of the obligation they are under, in common with all women, of whatever station, to perform domestic duties; and when you find that their principal domestic duties during the earlier part of the day consist in preparing for the two or three hours’ dissipation of the afternoon, you will easily see how little time there is left for attending to those matters in which home-comfort is a first consideration. And if you come to the men who revolve around these centres of silk, lace, French gloves, and Frangipani, you find them—well you find them worthy of their occupation. There are plenty of reasons for social intercourse, and in truth we want more of it in this community, but the social intercourse supplied in the process of “doing the block,” is exactly of the sort of which it is desirable to have as little as possible.

3-: From the column The Flaneur in Sydney, published in The Empire (Sydney, New South Wales, Australia) of Saturday 22^nd August 1868 [page 5, column 1]:

I have an immense admiration for the fair daughters of Sydney. They are the prettiest, wittiest, most utterly charming, bewilderingly fascinating, and doubtless perfectly amiable young ladies under the sun. To stroll on a sunny afternoon mid “the rosebud garden of girls” in George-street, is to anticipate the Heaven of the Moslem, and, to wish oneself a Turk. To contemplate the dress circle on a fashionable opera night is to experience what the Spectator terms “extreme cerebral excitement,” and to wish oneself—say, a Count. If I were Mr. Kendall I would write lyrics to them, after the chaste manner of Mr. Swinbourne, and dedicate them to the “siren with treacherous tresses,” as the “Maiden I meet at the corner.” If I were Mr. Montague Scott, I would paint them and be immortal. As I happen to be neither of those clever gentlemen, I will simply tell my fair friends in plain English, what I wish they would do for me. I wish they would curtail the proportions of those enormous trains with which they are in the habit of decoying bashful flaneurs into acts of ungallantry painful to commit and humiliating to remember. It is as difficult now-a-days to “do the block” without coming to grief with some fair but indignant dame, or rather with her costly appendage, as it would be to walk through nettles without getting stung. To ruin a dress in George street is almost as inevitable as to run against a porter in Whitechapel, and the Parthian glance which usually rewards that delicate feat, like the lady herself, must be seen to be appreciated.