You can use either this faucet adapter
https://www.bulkreefsupply.com/chrome-faucet-diverter-valve.html
or this angle stop valve, which attaches under the sink.
https://www.bulkreefsupply.com/john-guest-angle-stop-valve.html
Come to think of it, the fitting in your picture looks like a just plastic version of the angled stop valve and looks like it should attach to your cold water line under the cabinet above your kitchen sink. For your fitting, you could use teflon tape on the threads to stop the leaks that you’re experiencing but I would recommend a thread paste over teflon tape for this application.
https://www.amazon.com/Plasto-Joint-Plastic-Thread-Sealant-Temperature/dp/B004MYFP8C