The dragon breath is not an attack; your reaction is too late
The rules for Readying an action (PHB 193) say (emphases mine):
Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn, which lets you act using your reaction before the start of your next turn. First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger...When the trigger occurs, you [can] take your reaction right after the trigger finishes
You have defined your Readied Action as "I'd like to hold my action until the dragon aims it's dragon breath. The action I would like to ready is to attack the dragon." and said that "the gm states that he will use the dragon's breath attack on my friend, and aims the cone directly towards them."
These statements reveal that both you and your DM do not understand that the dragon's breath weapon is not an attack in the way that the rules define an attack and, in that sense, is not aimed at your friend.
Had the dragon actually made 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, what you call "aiming" would be Part 1, choosing a target. While a contentious issue, some DMs (myself included) would allow targeting to be a 'perceivable circumstance' and thus allow your Readied Action to trigger and resolve between Parts 1 and 2.
However, in the case of a breath weapon, this 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.
Thus, when your DM says that the dragon "aims the cone directly towards" your friend, that is not actually what happens. A more precise expression would be that the dragon chooses the location of its breath weapon to include the area in which your friend is located. And while you might like to react to that, this selection of area of effect and the resolution of the damage occur all at once, without the Parts that divide an attack. Since your reaction, by definition, occurs after the trigger, you are simply too late - by the time the dragon chooses the location of its breath, it has already breathed. You can certainly react to that, but by then your friend has already been roasted.