A sink gurgle noise, sometimes when the toilet is flushed, is trapped air in the pipes caused by a clogged vent pipe. The vent is the pipe that runs from your kitchen and bathroom drains and extends through the roof. This lets air flow behind draining water so it does not become air trapped. Remove the clog.
An example of a clogged vent would be similar to lifting a straw out of a glass of water with your thumb over the end. A clogged vent will hold water in the pipe until enough force is generated to suck air through a trap.
Usually the sinks, both bath and kitchen, connect to the vent about 16 inches above the toilet. When the toilet is flushed, it needs air to allow the water to drain down into the basement. Usually the air comes from the vent, but when it is clogged it will ‘suck’ air from other sinks, such as the kitchen.
Get a plumber’s snake, a manual one will usually do with about a 1/2 inch cable, and go on the roof and run it down the vent above the kitchen and bath. The clog should be in the first ten feet on a ranch home. Sometimes the vent makes a jog in the attic and here is where the clog usually forms. Prepare yourself to find an animal or animal nest. Typically, they are already dead, stinky too.
If you are not able to safely go on the roof contact a plumber and they will do this job. This problem is not uncommon in our area.