the cloud measures 20 x 20 x 20 km in size, and the particles are traveling at 30 km/s. The density of the dust cloud is 0.125 g / cubic meter.
That's about a million tonnes of debris there, with a total kinetic energy of 450PJ (about 100 megatonnes TNT equivalent, more or less). Each square metre of the front of the dust cloud has 2.5kg of dust behind it, giving a yield of ~1.125GJ/m2... that's about a quarter of a tonne of TNT equivalent. A ship with a 400m2 cross-section would be hit with the equivalent of a hundred tonnes of TNT, pasted all over its nose.
I'm not going to say that's not very much, but it is a fairly small proportion of the energy of the cloud. Ships are not going to be within 20km of each other, so that's a fair amount of overkill to throw at one ship, especially given that a) the acceleration mechanism is going to be very energetic (and therefore visible from far away), the cloud is going to occlude things behind it (and therefore be visible from far away) and is going relatively slowly compared to the vastness of space and so the target ship is going to have plenty of time to see it coming, and will not have to manoeuvre very hard to evade it.
What are some methods of defending against this weapon?
Given that its only working out as 2.5kg of mass per square metre of cross-sectional area of the ship, simple dumb armoring will be pretty effective. A sheet of aluminium 1m x 1m x 1mm weighs a little more than that, and might be reasonably expected to absorb a decent amount of the impact. A multilayer Whipple shield made of successive thin layers of suitable materials will protect the ship behind it quite effectively, and without an unreasonable mass penalty.
Obviously, this only works a limited number of times, but given the difficulty of generating the dust cloud in the first place and the good chance of the target seeing it and evading it, the shooter might not be able to hit the target enough times to take it out.
Detonating nuclear blasts wouldn't have any effect since there is no blast wave, right?
Well. A nuclear blast in space does generate a lot of radiation, and as MadScientist already observed you might reasonably expect the dust cloud to absorb a reasonable amount of that radiation which will vaporise some of it and deflect some of it. Remember the whole cloud doesn't have to be disrupted, only enough to limit the amount that hits the Whipple shielding (or other armor) that the target spacecraft has. If you hit the cloud far enough away, the deflected dust grains (and the gas expanding from the vaporised grains) will act like a blast wave and disrupt the cloud further.
However. Not all nukes are created equal! Behold the Casaba Howitzer, or even its non-militarized cousin the Orion drive pulse unit. These act as directional nuclear blasts in space, absorbing a substantial portion of the radiation emitted by the blast into a propellant disk which is turned into a dense plasma cloud that flies towards the target. This is very much the sort of blast that can disrupt the gas cloud, to the point where I feel that this might be the ideal defensive option.
Note that Orion propulsion units are potentially non-military devices (they may be nuclear warheads, but they're far from optimised to be weapons) and so could be used in defence by non-warships.
I can imagine high speed focused lasers, but it would take an enormous amount of power to actually destroy something with a laser - albeit at the zetajillion times you'd have to do it against a large sand cloud.
As with the nuclear radiation pulse above, you don't have to completely destroy a dust grain to reduce or eliminate its damaging capability. The dust cloud is optically thick, so a laser weapon might reasonably expect a large proportion of the energy of the beam to be absorbed. Some of the grains may be wholly vaporized, some will be partially vaporized which will deflect them, and for others it may be that the radiation pressure of the beam can provide a useful amount of deflection or slowing-down. Remember that deflected (or slowed) grains can collide with other parts of the cloud, further disrupting it. You're only limited by the cycle time of the laser, and the distance at which you can detect the dust cloud (which is likely to be far away).
I'd much prefer the Casaba Howitzer to a laser, though.
You might also be able to use laser driven lightsails... a series of featherweight sheets blown forwards by light pressure alone, but it seems difficult to get enough mass moving quickly enough to make a difference. A magnetic sail driven by a particle beam might be more effective, both in term of delivering mass and getting projectiles launched and accelerated quickly. By way of a bonus, these things might reasonably be carried for non-military purposes, which might be plot-relevant.
What are some methods of defending against this weapon?
Why not fight fire with fire? Generate your own high-speed clouds-o-doom, and fly behind them, relying on them to help sweep opposing clouds, drones, missiles and ships out of the way. Obviously you can't easily carry a million tonnes of grit around with you everywhere, but you should be able to do it at least once.
Honestly though, I'm not sure that this is a very useful weapon. It is difficult to produce, not practically portable, and is too easy to spot at extreme ranges and avoid. It also appears to be possible to defend against without undue difficulty.
Send your weapons engineers back to the drawing board.