1/4 sink cold water shut off valve doesn’t completely stop water
It is not uncommon for the stop-valves at the wall to fail (in various ways). The 1/4 turn valves generally seem to last longer than the traditional style valves, but they fail too.
Your only option now is to turn off the water at an upstream valve and replace the stop valve. Your home should have a main shut-off, or you may have to go out to where the water is metered and turn it off there.
I encourage you to go with a 1/4 turn valve again despite the trouble you have now; try to buy a quality valve, not the cheapest one. A plumbing shop (as opposed to a &quot;big-box&quot; hardware store) may be a better place to purchase a good brand.