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.