Maybe. Sort of.
No brindle gene has been found in horses and the coloration is vanishingly rare.
This is Dunbars Gold - and he sure looks like a brindle. Well, here's the rather interesting situation.
A breeder who owned a mare named Sharp One, also brindle, bred the mare to Dunbars Gold, with the intent (by the way, I am not typoing - the AQHA does not allow 's in registered names) of seeing if breeding a brindle to a brindle would give her a brindle foal.
Quarter Horses have to be DNA typed. When she sent in the sample, she was told that not only was Dunbars Gold not the sire of her foal but Sharp One was not the dam, and was told she had obviously sent in a sample from the wrong foal.
Snag is? She only had the one foal on the property. And Dunbars Gold's owner only had the one stallion. There was no way the sample could have been mixed.
So, they re-tested both the sire and the dam and when they did a blood type on Dunbars Gold they found he was a mare! No Y chromosome to be found. Obviously, this also could not be accurate, because he was siring healthy foals. So they went back to hair typing - and discovered that Dunbars Gold was, in fact, a chimera - twins that fused in the womb.
Aha. Lets test the mare now. And her other foals for good measure. They discovered that the foal they were testing was showing as unrelated to both the mare and his older sister. Explanation? Sharp One is also a chimera, and one of her ovaries is from one cell line and the other from the other...
So, are all brindle horses chimeras? Most are. In 2014 an inheritable form of brindle was discovered in a single family of horses, but it turned out to be a rather nasty disease that resulted in male fetuses being non-viable and the surviving females having abnormalities of the teeth, hooves and eyes, and it also causes areas of hairlessness. These brindles had Incontinentia Pigmenti (there's a similar condition in humans).
In other words? There are no true brindle horses, but if you want to put one in your story, go ahead. (And if you want a cool trait for a fantasy horse breed that doesn't really exist in the real world, how about an entire breed of brindles?)
Image source: Kersti Nebelsiek via Wikimedia Commons.