Dragon Breath is not an attack and does not "target someone"
Unlike the currently accepted answer, I do believe that targeting is at least sometimes perceivable. It is specifically the character's perception of this targeting that allows such features as the Protection Fighting Style, the Goblin Boss's Redirect Attack, the Mastermind Rogue's Misdirection ability, and the redirection of attacks permitted by the Mounted Combatant Feat to function; they simply could not work as written without it.
However, in this case we are not talking about the targeting of an attack because a dragon's breath weapon is not an attack in the way that the rules define an attack and, in that sense, the dragon is not "targeting someone with their breath weapon".
Were the dragon actually making an attack, such as a claw, bite, or tail strike, that attack would have three steps:
- Choose a target.
- Determine modifiers.
- Resolve the attack.
In this case, Part 1 would be choosing a target. While a contentious issue, some DMs would allow targeting to be a 'perceivable circumstance' and would thus allow a Readied Action to trigger and resolve between Parts 1 and 2.
However, in the case of a breath weapon, it is not an attack and does not have these three parts. In 5e, an attack is, and is only, something that requires an attack roll (or is specifically called an attack if no attack roll is made). The dragon's breath weapon does not meet either of these conditions. As an example, here is the fire breath from an adult red dragon.
Fire Breath (Recharge 5–6). The dragon exhales fire in a 60-foot cone. Each creature in that area must make a DC 21 Dexterity saving throw, taking 63 (18d6) fire damage on a failed save, or half as much damage on a successful one.
Note that there is no mention made of an attack roll, an attack, or of targeting an opponent. Said dragon could quite deliberately roast creatures under the effects of a sanctuary spell without ever having to make a save.
Since the dragon is merely electing an area over which to breathe, rather than choosing individuals to breath upon, there is no Making an Attack to interrupt.
By the time your caster recognizes what general area the dragon has chosen, and realizes that said area includes an ally or two, it has already breathed, and the wall of stone would follow thereafter and would be too late.