If you only have access to the inside of the duct, and then only as far as you can reach, you won't be able to make a huge difference, but you might be able to reduce it a little.
The sound is transmitted in mainly two ways, some is conducted through the metal of the duct and some is reflected around the inside of the duct until it reaches you. You can reduce both a little, but remember that:
- You're breathing the air that comes through. You don't want to use materials like fiberglass.
- The HVAC system is tuned to the volume of air that it's moving. Also, temperature is based on how much air flows into the room. You can't significantly reduce the air flow without affecting heating/cooling in the room, and if you reduce total air flow in the house too much, it puts a load on the HVAC system that can make it inefficient or even cause premature wear.
Conducted Sound
You can reduce sound conducted through the metal by dampening it. This is usually done by attaching a deadening material to the duct. It is typically a very soft rubber or gel-like material. The vibration is absorbed by the material and dissipated as heat. You don't need to line the whole duct, just a relatively small area. One manufacturer has a demonstration video where they attach one piece a few inches square to a cymbal and the sound becomes a thud.
There are many manufacturers, and I haven't personally used any of the products, but here's an example from Amazon that has decent reviews:
This type is a soft butyl rubber sheet that absorbs the vibration. One side has adhesive. The other has a patterned foil. You use a roller to press the material against the duct for a good bond. The foil is just used so you can see where you've already rolled (the pattern flattens out). Inside a duct, the foil would also serve to keep dust from sticking to the soft rubber. This stuff isn't thick enough to materially change the the air flow through the duct.
I would clean the inside surface of the duct as far as you can reach. Then put the sound deadener on the bottom and sides of the duct to the distance you can reach. You can easily cut it to the size you need with scissors. You don't need to go edge to edge on each surface, just cover a good portion of it.
Reflected Sound
For the sound reflected through the duct, you can't really block it, but you can absorb some of it to reduce what's bouncing around inside the duct. In tests of materials for sound absorbing panels, ordinary cloth towels turned out to be on par with fancy materials designed for sound reduction.
Based on how far you can reach into the duct, fold something like a bath towel so that it's almost the width of the bottom of the duct and as long as you can reach. You want as many layers as you can fit in no more than maybe 1/4 the height of the duct so you don't affect the air flow too much. Stitch the leading and trailing edges so the air doesn't blow the folds open. Then lay it on the bottom of the duct.
An alternative to the towel would be polyester fiberfill. This is the stuff sold in fabric stores or places like Amazon or Walmart for stuffing pillows. It's sometimes used to fill the inside of a ported speaker to reduce sound reflected internally.
It doesn't reduce air flow all that much because it's mostly open space. You could take a bag or two of it (enough to fill the cross section of the duct to a depth of a few feet), try to keep it in an intact wad, and stuff it into the duct. Push it in just enough so it doesn't interfere with the register. If you find that it blocks too much air, pull some of it out.
You could also stuff some fiberfill in the duct at the tenant's end.
