Since you explicitly state in the comments above that you don't feel you can "separate the gameplay experience from its mechanic" and that you expect to get "answers from different backgrounds", let me provide the (almost) purely mechanical answer I was alluding to in my own comments. I'll start by noting that I've never played a game using either Fate/Fudge or the AGE system, but I believe that your question, as phrased, is technically answerable without such experience. All it takes is some math and creativity.
(I'm not saying that this is necessarily a good answer — although I did honestly try to make it as good as it could be, given what it is — or that it necessarily solves the OP's actual problem. But I believe it does technically answer the question.)
So, without necessarily changing dice colors or using any other dice than just vanilla Fudge/Fate ones, how could I reproduce this type of mechanic? (For now, just ignore the point-buy mechanic since the Fudge/Fate die is limited. Let's assume when doubles happens it will generate just one single point).
So you want something that, using a 4dF roll as in Fate/Fudge, generates "bonus points" with a similar frequency as the AGE stunts mechanic using a 3d6 roll generates (groups of 1–6) stunt points. So what are the key features of the AGE stunts generation system? Let me list a few that I'd consider relevant:
It requires a successful attack. That's something that carries over to Fate easily enough, but it's worth keeping in mind.
When an attack does succeed, it triggers pretty often. Ignoring the success requirement, the probability of rolling a double (or a triple) on 3d6 is exactly 4 / 9 ≈ 44.4%. The success requirement changes this a bit, since the probability of a double varies depending on the total sum rolled (in particular, it's 100% for rolls less than 6 = 1 + 2 + 3 or greater than 15 = 4 + 5 + 6), but it never goes below 40%:
Except for the success requirement, it's not strongly correlated with the sum of the roll. In fact, the AGE mechanic is symmetric in the sense that the probability of getting a double on a roll of 3 + X is exactly the same as on a roll of 18 − X:
...but it always triggers when you roll really high. As the chart above shows, you always get one or more (in fact, four or more) stunts when you roll 16 or higher, as long as that counts as a success (which it usually should). You do also always get at least one stunt point when you roll 5 or lower, but such rolls rarely succeed against any meaningful opposition.
With that in mind, let's see what kind of mechanic might recreate those features using a 4dF roll instead of 3d6.
As a quick first attempt, rolling two or more zeros on 4dF has an average probability of 40.7% (ignoring the success requirement) which is pretty close to 44.4%, and the probabilities are symmetric like in the Dragon Age mechanic. But it fails criterion 4 above, since you can never roll a double zero when the sum is +3 or more (or −3 or less).
Playing around a bit more, it turns out that we can get an exact 44.4% occurrence rate by marking one of the 4dF as special (just like in Dragon Age!) and awarding a "bonus point" whenever exactly one of the normal dice matches the special die. This trigger rule is also symmetric with respect to the sum of the roll, but it also turns out to have the same problem as the previous rule: you can never get a bonus point on a roll of +3 or more (or −3 or less).
However, approaching the problem from the opposite angle and starting with the requirement that a natural +4 roll should always earn a bonus point, it turns out that awarding bonus points on every successful attack roll with three or more matching dice fits all the criteria fairly well:
Ignoring the success requirement, triples occur with a probability of 1 / 3 ≈ 33.3%. This is noticeably less than in the original Dragon Age mechanic, but perhaps not unreasonably so.
As with the original mechanic, the conditional probabilities are symmetric with respect to the sum of the roll and are highest for very high (and very low) rolls:
The graph of the probability of getting a "bonus point" on a successful roll for various target thresholds looks quite similar to the original. In particular, the probability of getting a bonus point on a success is always at least 27%:
So, assuming that the somewhat lower average frequency isn't a showstopper, I feel like this triple-on-4dF mechanic provides a fairly close match the the original double-on-3d6 trigger mechanic.
