We report a size-controllable and low-cost fabrication method of graphene quantum dots (GQDs) using a thermal plasma jet. A carbon atomic beam was generated by injecting a large amount (2.5 L/min) of ethylene gas continuously into Ar plasma. The beam was then flowed through a carbon tube (5-20 cm in length) attached to the anode and then dispersed into a chamber. Carbon materials including GQDs were made by a gas phase collision reaction. The production rate of carbon soot was 40 g/h for a 2.5 L/min injection rate. Almost all of the carbon soot dispersed in ethanol by sonication, while isolated GQDs were dispersed in ethanol by stirring with a stirring rod. The weight percent of GQDs in carbon soot, based on the amount extracted in ethanol, was about 10%. This means that the production rate of GQDs was about 4 g/h. The average size of GQDs, with a relatively narrow size distribution, was controlled by varying the length of the carbon tube attached. It was about 10, 14, and 19 nm when the length was 5, 10, and 20 cm, respectively. The electric structure based on the photoluminescence data of our GQDs had a singlet ground state and was in good agreement with that of carbyne. Our GQDs will disperse in organic solvents such as toluene, but not in water. The dispersion properties also support that our GQDs have carbyne-like edges. We proposed that the PL peaks observed can be attributed to electronic transitions between energy levels of the GQDs having carbyne-like edges.