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I thought of a concept for a dragon or similarly gigantic reptile with a special adaptation, in which they’re born with scales made almost entirely of carbon. Then, over an incredibly long life cycle, the reptile would shed these scales, and replace them with diamond scales that they had formed inside their body using carbon. I formed this concept off the fact that life on earth is carbon-based.

So, my question is, would a large organism be able to produce enough heat & pressure inside its body to make diamonds?

Is the story strictly limited to diamond? If not you may consider silicon carbide (carborundum or moissanite). It can be synthesized at dragon-achievable temperature of 1100K. It is somewhat less hard than diamond but it's not flammable and tougher.

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.

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? 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).

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.

Since Chemistry is a marvelous science, I think it explainable, from a story telling stand point, for a dragon to grow diamond scales.

In chemistry, a catalyst makes reactions happen at lower energy levels and is not used up by the reaction. And, in semiconductor growth systems, crystals are grown using complex molecules called metal organics. And, since many crystals grown for semiconductor wafers share the same Zincblende crystal structure as diamond. It is reasonable to speculate that a fantastic creature could convert raw carbon into one of carbon’s crystalline forms using a complex organic chemistry processes and catalytic chemistry.

Is there an similar technology today that grows diamond crystals at atmospheric pressures and dragon temperatures, I don’t know. But, is it conceivable that we will figure it out? Yes, I think it is.

And, the scales need not be solid, but could incorporate a structure that makes the scale more resilient and less susceptible to shock. Like the scales could be honeycombed so when they are hit, the scales flex into the their internal void rather than cracking and shearing.

An organism could hypothetically gain diamond scales by consuming extant diamonds and incorporating them into its body. Such a creature might require an unorthodox method of energy generation because — depending on the environment and scarcity of diamond — it could have to spend a significant portion of its time foraging for diamonds in order to accumulate a meaningful quantity. Gaining energy from minerals encountered while foraging seems an interesting possibility.

See nudibranchs for an example of a real Earth organism that consumes other creatures which contain toxic stinging cells. After consumption it moves those cells through its specialized digestive system into its extremities where it then uses those "stolen" cells for its own defense.