Possibly, but you have to keep doing it. kicking up dust into the atmosphere tends to have significant short-term impact on the global climate (see e.g. volcanic winters: https://en.wikipedia.org/wiki/Volcanic_winter). Then the dust settles shortly and everything returns to normal.
But volcanic eruptions that cause a winter like that tend to be quite a bit more powerful than your average nuclear weapon. More importantly, not all dust is created equal. Sulphur containing compounds tend to have the biggest cooling effect. One neat (in a terrifying sort of way) geo-engineering idea is continuously injecting large amounts of sulphur dioxide into the stratosphere.
But volcanic eruptions that cause a winter like that tend to be quite a bit more powerful than your average nuclear weapon. More importantly, not all dust is created equal. Sulphur containing compounds tend to have the biggest cooling effect. One neat (in a terrifying sort of way) geo-engineering idea is continuously injecting large amounts of sulphur dioxide into the stratosphere.