Maybe yes, but not through that mechanism.
You don't necessarily need heat and pressure to make diamonds, and those are not usually things that we biological squidgey things use to grow things like horns, nails and scales.
I think there are two ways currently used for creating artificial diamonds on an industrial scale, one being the one you mention (High-pressure, high-temperature synthetic diamonds).
But the more interesting one is chemical vapor deposition. Now, this takes place in a vacuum and blahblah. So it, too, is not really a very good way of growing scales.
But at its root, it's basically just accretion. Which is how most biological stuff grows. As @MikeNichols puts it in a comment: "Living organisms don't need high pressure or high temperature to do chemistry because they have enzymes." He gives this relevant link: Could a living creature produce graphene? which is a different molecular organization, but has the similarity that they're both just ordered arrangements of carbon atoms.
Is it feasible that somewhere in the universe, some carbon-based life-form has developed a way to perfectly accrete carbon atoms onto each other, inside its body? Sure. Why not?
Is it likely, or advantageous, that they wold use diamond, for armor? No, it seems like it'd be a lot of work for "diamond perfection", when "good enough" scales are way easier to grow, and tougher (less brittle).
But nothing sparkles like diamond, and if you're trying to attract a mate as you fly up from the dark ground into the sunlight of dawn... yeah, diamond has advantages :)
As protective scales, diamonds would leave a lot to be desired, though. They fracture rather than flexing. When they break, they can split or shatter, rather than merely bending, wearing, or chipping. So that would need to be addressed, whether with logic, handwaving, or lampshading. See Is diamond armor better than traditional armor? and possibly Is a diamond sword feasible? for more info on that part, though.
Another possible limit is the availability of carbon in the body, which is covered here: What size would a diamond made from a human be?
Another option, other than growing the scales through accretion, is something like caddis-fly larvae, where the dragon clads itself in diamond for protection or mating. This would require an environment where diamonds are very common, however, so this armor would have little monetary value.
UNLESS... the dragons dived for them in the cones of volcanos or something, where normal humans could not go.
This would create something like Dragon Skin™️, except that , as @VilleNiemi points out in a comment, its "scales" were silicon carbide rather than diamond.
Can you please define how long you would like for this to take? What else is the dragon made of? For example, what is the carbon being crushed between to provide the pressure? – Muuski – 2019-05-17T16:40:22.793
@Muuski I’m thinking that the dragon takes around fifty years to reach full sexual maturity. As for what it’s made of, it’s more or less a normal reptile, albeit very large. So, maybe some sort of specialized system of muscular chambers? I’m not sure, so I’m looking for input on what systems could achieve this. – Cobbington – 2019-05-17T18:45:33.947
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Possible duplicate of What is the strongest glass that can be formed from nature?
– JBH – 2019-05-17T19:37:17.613Even if you disagree that this question isn't a duplicate of the other, @Green's answer to the other question is the right answer for this question. (I'm kinda of the opinion if Q#1's best answer is A and Q#2's best answer is A then Q#1 = Q#2 no matter how they're asked. But that might deserve a meta discussion.) – JBH – 2019-05-17T19:39:36.420
@JBH you just broken my brain... – Morris The Cat – 2019-05-17T19:40:31.803
@MorrisTheCat, um... anything worth doing is worth doing well? – JBH – 2019-05-17T19:43:08.960
Great find, @JBH, though only related: that asks "what's the strongest non-crystalline glass a human can artificially form from natural sources", this asks "how would crystalline diamond form in a biological creature?" - they have related concerns, and OP should read that question: but Green's answer to that question does not address, let alone answer, this question, and nor do any of the other answers. – Dewi Morgan – 2019-05-17T19:45:53.780
@DewiMorgan, yeah, but the answer to the OP's question is "no." I took the liberty of assuming what the OP meant. I could be wrong, which is why the voting process exists. On the other hand, Green's answer is a biologically-plausible solution where diamond is not. – JBH – 2019-05-17T19:47:47.387
Another related concern, that might very significantly limit this answer if not handwaved away: https://worldbuilding.stackexchange.com/questions/98455/what-size-would-a-diamond-made-from-a-human-be
– Dewi Morgan – 2019-05-17T19:48:36.4431@JBH I've given the answer "maybe yes", so I think "no" is an opinion, not a proven truth, and certainly nothing in Green's answer gives "no" to this question. If you feel you know the answer to a question, merging it with an unrelated question is rarely going to be the right way to write an answer to argue your point :D – Dewi Morgan – 2019-05-17T19:51:45.027
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This question is not a duplicate. It is closely related to this question though: Could a living creature produce graphene? They differ only in which allotrope of carbon they want to produce but the answers are basically the same. Living organisms don't need high pressure or high temperature to do chemistry because they have enzymes.
– Mike Nichols – 2019-05-17T20:01:46.640I don't know avout reptiles but if a crab evolves a diamond carapace they are contractually obliged to sing the song Shiny from Moana. – Renan – 2019-05-23T14:52:28.797