How do I get rid of the sulphur smell from cold water only in one sink?

To further diagnose this, get a glass and fill it with cold water, then smell it away from the sink, if it stinks, you most likely have an anaerobic bacteria colony in the piping near that particular sink. It smells like sulfur. I've had that problem in my house (well water) repeatedly. I finally think I got it all and haven't had the problem in a few years now.

The fix is bleach. It doesn't need to be full strength, about 20% solution (1 part bleach 4 parts water). I bought a small pump and a new trash can for the purpose. Fill the clean, new trash can about 1/3 full and put in a gallon of bleach. Then turn off the water to the house. connect the pump to an outdoor hose bib, You'll likely need a female-female hose adapter. Then turn on the pump and run cold water until you can smell bleach. Then turn the pump off and let the bleach solution set for a few hours. Since you have only one sink affected, I wouldn't bother with other fixtures, but that's up to you.

After a few hours, turn the water back on for the house and run the cold water. Don't be surprised if you see a bunch of heavily discolored, even black, water come out of the tap. In extreme cases you might have to remove and clean the aerator. If you get heavily discolored water, that's confirmation that you had anaerobic bacteria colony.

Let the water run until it clears up and you don't smell bleach anymore. You might get a bit of bleach smell for a few days, but it will go away.

Don't run any hot water when doing this because you'll pull the bleach solution in the HW tank and it takes forever for it to flush out (don't ask me how I know this).

A few caveats: If your hose bib has any sort of backflow prevention this setup won't work and you'll have to find some other way of introducing the bleach solution into your plumbing.

When turning on the water again, just to be safe, run all the cold water taps at some time in case the bleach solution went to some of the branch lines.

Again, don't run any hot water taps during this process.

If you have a hot water recirc pump, turn it off. They don't like being without pressure.

Also, this is just a guess on my part on what's causing your problem, but it sure sounds a lot like what I experienced. If you aren't comfortable doing the above procedure, you should probably get a plumber to help.