Would a change in the initial concentration of SCN- affect the equilibrium constant of the reaction (Fe3+ (aq) + SCN– (aq) ↔ FeSCN2+ (aq)) ?

Changing the concentration of reactants or products does NOT change the value of the equilibrium constant. After all, it's a constant. Changing temperature WILL change K but concentration, pressure, volume, will not. Note that changing volume, pressure or concentration of reactants and/or products will cause the reaction to shift and change the value for products and/or reactants for some reaction but the value of K remains the same except when T is change.