No, cheaper gas is not lower quality. In fact, the price of gasoline has very little to do with its quality. The main difference between gas prices is the taxes that are added to them.
The octane rating is a measure of a fuel's ability to resist knocking or pinging during combustion, caused by the air/fuel mixture detonating prematurely in the engine. In the U.S., unleaded gasoline typically has octane ratings of 87.
The higher the octane rating, the more resistant the fuel is to knocking and pinging. However, this does not mean that higher-octane fuels are automatically better for your car. Unless your car is designed for high-octane fuel (like some luxury and performance vehicles), using it will not improve your car's performance or gas mileage. In fact, it could actually damage your car's engine