Somewhat counterintuitive, but Deimos would be a better choice than Phobos. Neither would be easy.
Phobos has an orbital velocity of nearly 7,700 kph. Deimos, a bit better at about 4,860.
To shoot dust off either moon, that's roughly how fast it would need to leave the moon. It's not just escaping the moon's gravity but escaping the orbit, and there's an interesting quirk about orbits. It takes more energy to land from a near orbit than from a far one.
Idea explained here:
A distant asteroid orbiting Mars (don't think there are any) and an accurate cannon would be the least energy intensive.
The clouds they discuss are (probably) dry ice clouds, though the article doesn't say but the high altitude suggests that's what they are.
The article isn't very specific on how the clouds warm the planet, other than slowing down the Hadley cell rotation. I share Michael Walsby's concern that clouds reflect heat as well as trap heat. I'm not sure how much warming you could get by creating more clouds, and the slowing of the Hadley cell suggests that the poles might get colder, not warmer - but I've not read the study in detail, so I'm skeptical but not certain.
But I don't see why an approach like this couldn't be done. Not with current technology and planned space trips, but in the not too distant future, equiptment to turn parts of the surface of a moon, like Deimos into fine powder and a method to shoot that fine powder onto Mars, several tons a day. That wouldn't be cheap, but it's not entirely out of reach. It could probably be done, in maybe 30-50 years.
The big question is why do it at all? Warming Mars by a dozen or more degrees doesn't accomplish all that much. It doesn't make it all that more livable for any astronauts who go there. Mars' near vacuum is the real problem, and it's nasty and toxic surface dust, and it's lack of nitrogen.
But to answer your question, yes, I think it probably could be done, but I don't think it will be done in the near future because it's too high cost for basically no benefit. Creating clouds on Mars wouldn't accomplish very much and fluctuations in temperature even as much as 10 or 15 degrees wouldn't be a major factor in making Mars more habitable, unless these clouds and the weakening of Mars' Hadley cell would prevent dust storms, in which case, that would be a positive. Martian dust storms would be a pain in the but to astronauts there and Martian dust storms have been observed to be planet wide and last longer than a year - so, avoiding those would be of interest.
In the long run, maybe 200-300 years in the future if all goes well, and if I was to speculate a bit, then a plan to terraform mars might get under way, but lots of study will be done first. It's possible that shooting dust onto the Martian atmosphere and creating high atmosphere CO2 clouds might be part of that plan.