We currently live in a 5 year old home that has three bathrooms. Now, the problem persists in our Master Bath. After you use any water source in the bathroom, the toilet makes a rushing air noise (for a second or two) every 5 minutes. It subsides after about 45 minutes.

I looked in the toilet tank and waited to see what it does. When it makes the noise it shoots a squirt of water out the side of the pump thing (not sure of the proper term) through a small hole?

Is this an $$$ fix? Please help!

It sounds like you have a bad flapper or the tank water level is too high.

Check the "critical level" Mark at least an inch over the rim of the white overflow tube.

The water level normally falls about 1 inch below the rim of the overflow tube. Lower levels don't allow the float ball to raise high enough to close the ball cock; a water level that is too high will continuously run, wasting water and forcing you to listen to the running water all day

Feel around the flapper flat edge. If it's wavy or you feel rough spots replace the flapper,
The linkage between the flapper and the flush lever should have 1/4" play when the tank is filled. Also, check the small tube coming out of the top of the ballcock and make sure it's connected to the white overflow tube and draining water in to the tube.

Also you can check if your flapper is leaking place some food coloring in the toilet tank, after an hour or so check if dye showed up in the toilet BOWLS then you will know for sure that the flapper valve is defective and needs to be replaced, also see the images below and tell me which fill valve do you have inside your water tank, good luck.

John