[john70t]

From the first pic:
The handle set screw is always either located at the direct top (usually a plastic Blue/Red tab pried out), or it is hidden on the side.
The pic above indicates this one is on the side.

Put a rag in the drain

Unscrew that but not all the way.
-Handle will pop off.
-The square base enclosure will then lift off.
-What is left will be a valve attached to the top of the counter with some sort of surface washer and nut.

If you want to remove that:
-Shut the water off at the 1/4 turn under the cabinet.

-Be careful about this.
You do not want to bend pipes at the wall.
Put a rag behind and spray it with WD-40 from all sides. Bucket and towels under.
Wait. Go easy when trying to turn. May need pliers or a rag.
Do not force. Go easy.
It should eventually shut off all water with enough massage and patience.
Some have to be worked with water ON to wash out calcium buildup.

-If the sink 1/4 turn valve doesn't stop the main flow, you'll have to shut off the whole house at the meter and drain everything from a basement sink.
No water for anything.
Try to avoid that.

-Some 1/4 turn drips can be fixed by slightly tightening up the base nut underneath or the set screw front of the handle.
Wiggle. Wiggle. Gentle. Gentle. Easy on those threads.
Observe.