The outdoors faucet is running but the hose does not have water coming out, unless squirted out for 3 days. Will this cause an increase in the water bill?

You will be charged for any water used on your side of the meter.
If the faucet is on and no water come out the hose, you are not using any water.
See if indicator on Water Meter is Spinning, if you have that type.

You are asking this for a reason. Would you care to share? High water bill?
To answer your question, If the hose isn't running there will be no charge on your bill. Regards, Tom