Yes, with a few caveats.
First of, yes, you can use any mount you can possibly imagine and create a story around it for yourself. That is your head-canon and no-one can tell you otherwise.
However, how other people react to your mount is up to the type of people and characters they are. And how “high magic” they like to keep their roleplay at.
If you are playing among friends who keep to a high-fantasy theme, none of them is going to have a problem with your mount. However, a community of gritty survivors trying to eck a living in the mountains, or the ordinary townsfolk of…let’s say Stormwind may prefer to stick to more “down-to-earth” themes. To them, that Dragon may be disruptive. Too high-magic.
It also depends on how you sell it.
If you land in the middle of the street, with little to no description, you may give an image that you’re trying to play a special snowflake with a unique mount.
However, if you describe with loving care, in detail, on how you land and gracefully climb off elegantly like a disney princess, it shows you’re putting effort behind your concept, and that is harder to shrug off - it shows you’re serious.
But still, it depends on the people you play with, and how open-minded they are for the high-fantasy themes of the game. And what is the “power-level” of people you interact with. We have a whole spectrum of Archmages and Archdruids to low-magic mercenaries and bandits in the game. You do not want to appear out-of-place for the narative, just like you probably won’t bring a Lightforged Draenei on their shining steed to a Warlock Cove meeting.
That is my few cents. And below an example of my own use of an exotic mount:
On one character, I used to use Ashen Pandaren Phoenix (MoP Challenge Mode Silver mount) and claim it was a little songbird from Krasarang Wilds that my arcanist mutated using energies of the Emerald Dream into their current, ride-able form.
With the above example, only a select few people of my Secretive Magic Cove of Magic Users ever saw my character using Ashen Pandaren Phoenix. We played high-magic, skilled if secretive wizards. We had dreadsteeds and other weird things. Why not strange dream-songbirds?
I never showed it in public, in consideration to lower-magic roleplayers, but also for In-Character reasons. That thing would’ve been an abomination to any Druid, and suggest of quite shifty, dark Arcane research to any other Wizard.
The character didn’t want such attention, so they only called for it when they were far from civilization, away from town, and used it to cross vast distances by flight - by always landing in the wilderness, outside the line of sight of the city walls. That is how they travelled.