A boil-off gas supply device comprising: a storage tank configured to store a liquefied gas; a first compression mechanism configured to suck in a boil-off gas of the liquefied gas stored in the storage tank and compress the sucked boil-off gas; a second compression mechanism configured to compress the boil-off gas after being compressed by the first compression mechanism; a discharge path in which the boil-off gas discharged from the second compression mechanism flows; a first drive source configured to drive the first compression mechanism; a second drive source that is different from the first drive source and configured to drive the second compression mechanism; a return part configured to re-liquefy the boil-off gas discharged from the second compression mechanism and return the re-liquefied boil-off gas to the storage tank; and a first discharge-side pressure sensing part configured to detect a pressure of the boil-off gas on a discharge side of the first compression mechanism and on a suction side of the second compression mechanism; a suction-side pressure sensing part configured to detect a pressure of the boil-off gas on a suction side of the first compression mechanism; wherein the boil-off gas discharged from the first compression mechanism is supplyable to a demander side, and a second supply outlet of the boil-off gas is provided in a branch path branched from a connection path that connects between the first compression mechanism and the second compression mechanism, characterized by a first spillback mechanism configured to return a part of the boil-off gas discharged from the first compression mechanism to a suction side of the first compression mechanism; and a capacity control part configured to control the first spillback mechanism and/or a capacity of the first compression mechanism; wherein the capacity control part is configured to control the first spillback mechanism and/or the capacity of the first compression mechanism so as to decrease an amount of gas compressed by the first compression mechanism when a pressure detected by the suction-side pressure sensing part is lower than or equal to a second threshold value, and the capacity control part is configured to control the first spillback mechanism and/or the capacity of the first compression mechanism so that a detected value of the first discharge-side pressure sensing part comes to be a target pressure that is set in accordance with an amount of demand of the boil-off gas when the pressure detected by the suction-side pressure sensing part is higher than the second threshold value. a storage tank configured to store a liquefied gas; a first compression mechanism configured to suck in a boil-off gas of the liquefied gas stored in the storage tank and compress the sucked boil-off gas; a second compression mechanism configured to compress the boil-off gas after being compressed by the first compression mechanism; a discharge path in which the boil-off gas discharged from the second compression mechanism flows; a first drive source configured to drive the first compression mechanism; a second drive source that is different from the first drive source and configured to drive the second compression mechanism; a return part configured to re-liquefy the boil-off gas discharged from the second compression mechanism and return the re-liquefied boil-off gas to the storage tank; and a first discharge-side pressure sensing part configured to detect a pressure of the boil-off gas on a discharge side of the first compression mechanism and on a suction side of the second compression mechanism; a suction-side pressure sensing part configured to detect a pressure of the boil-off gas on a suction side of the first compression mechanism; wherein the boil-off gas discharged from the first compression mechanism is supplyable to a demander side, and a second supply outlet of the boil-off gas is provided in a branch path branched from a connection path that connects between the first compression mechanism and the second compression mechanism, characterized by a first spillback mechanism configured to return a part of the boil-off gas discharged from the first compression mechanism to a suction side of the first compression mechanism; and a capacity control part configured to control the first spillback mechanism and/or a capacity of the first compression mechanism; wherein the capacity control part is configured to control the first spillback mechanism and/or the capacity of the first compression mechanism so as to decrease an amount of gas compressed by the first compression mechanism when a pressure detected by the suction-side pressure sensing part is lower than or equal to a second threshold value, and the capacity control part is configured to control the first spillback mechanism and/or the capacity of the first compression mechanism so that a detected value of the first discharge-side pressure sensing part comes to be a target pressure that is set in accordance with an amount of demand of the boil-off gas when the pressure detected by the suction-side pressure sensing part is higher than the second threshold value.