How many generations does it take for one species to become a new one?
Your question is like asking about the color spectrum. A wavelength of 530nm is green; a wavelength of 580nm is yellow. So let's say it takes 60nm of wavelength change to go from green to yellow. What color does a wavelength of 579nm have, green or yellow? If it's green, then does the color shift in a single nm difference of wavelengths? If it's yellow, what color is 578nm, etc?

The answer in this case is, 579nm is yellow; so is 578nm. 570nm is more ambiguous; some people might say it's yellow, others think it's greenish-yellow, in some contexts it might look clearly green. And so on along the spectrum; the reality is that there is no hard boundary between "green" and "yellow". There are colors that are clearly green, colors that are clearly yellow, and there is a difference between those two, but there are also plenty of intermediates that are a bit like green and a bit like yellow, and those intermediates are what you go through when you go from green to yellow. There is no definite point everybody can agree on where a single shift happens.

It is the same thing with species. A species is a group of organisms that interbreed to produce fertile offspring, but "interbreeding to produce fertile offspring" isn't a on/off thing. Organisms are more or less fertile. What if organisms from two groups interbreed and produce offspring, and only half of those offspring are fertile? Or they have half as many offspring as couples within the two groups have?

The reason species are an important concept, and can be distinguished visually as being like each other and unlike members of other species, is that within a species there is widespread sharing of genes over the generation, as individuals breed with each other, but they don't share genes with other species. Meaning species will evolve as a unit; changes that happen within a species will spread between individuals of that species, but won't spread outside of it.

But because sharing genes isn't an on/off thing, you could have 100% of genes being shared from one population to another (then they're clearly the same species), or 0% (then they're clearly not), or 90%, or 70%, or 10%... That's where you find the fuzzy boundaries. The races, strains, varieties, sub-species, all those different words to describe populations that are different but aren't outright different species. And the debates on whether different groups are different species or not in the first place, in the cases where it's really ambiguous.

And the same thing is true of evolutionary change in general. You don't have a mammalian carnivore that isn't a cat, and then bam! a cat. You have a mammalian carnivore that isn't a cat (or dog, or weasel... but kind of like any of those in various ways. Like generic carnivore-ish), then a mammalian carnivore that's a bit more like a cat or a dog (and another one next to it that's a bit more like a weasel), then a mammalian carnivore that's even more like a cat (and others that are even more like a dog, and even more like a weasel), then a mammalian carnivore that's clearly feline but isn't a cat, or a lion, or a tiger; it's a different species of feline that doesn't exist today, and also looks more generic-feline-ish than any modern feline. Then there's a feline that's more like a cat (and others around it that are more like a lion, and a tiger), but also clearly isn't a modern cat. Then there are felines that are really like cats but a bit different from modern ones and it's hard to tell, maybe we should extend the definition of "cat" to include them? And then bam! a cat. But that cat isn't clearly a Jungle Cat, or a European Wildcat, or a domestic cat, or any modern species; it's its own "archaic cat" species. And then there are cats that are more like Jungle cats, or more like domestic house cats...

Also don't confuse the creation of new species with evolutionary change in general. Over the long term they are related, because once two groups are different species then they will change through evolution in independent directions, and they will end up looking different from each other as well as from their common ancestor. But over the short term you can have speciation with very little morphological or genetic change (you just need two groups that used to interbreed, to no longer do so), or morphological/genetic change without speciation (if all the change happens within one group).

Both speciation and genetic/morphological change can happen within a single generation, but usually happen over much longer timescales. But there is no hard rule for how long that is; as far as speciation goes, if groups aren't prevented from interbreeding in some way then they will keep breeding with each other and can stay a single species indefinitely (whether individual from generation 10,000 could interbreed with individual from generation 1 is a different question, but it's not like it's a situation that happens). As far as evolutionary change goes, natural selection means species change to adapt to their environment. Once they're adapted to it, if it happens their environment never changes then natural selection will push them to stay the same as they are, again indefinitely. In the real world most environments change over time so organisms do change more often than not, but just to illustrate that the timelines are a product of circumstance, not an intrinsic feature of evolution itself.