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Yes, as stated previously, temperature is the only factor that can alter K in a reaction at equilibrium. For future reference I believe we also have to know how pressure/concentration alter the direction of the reaction but these changes to not affect K.
The reaction quotient depends on the concentrations/pressures of the reactants and products. For reactions not at equilibrium, these values are constantly changing and thus Q is nonconstant. Even for reactions at equilibrium, changes in the concentrations/pressures (adding/removing more reactants/products, compressing the reaction volume, etc.) will result in a change to Q as well.
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