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ICE tables are almost always used to find the equilibrium concentrations, hence the E standing for Equilibrium in ICE (Initial, Change, Equilibrium). Questions involving Q would be giving you initial concentrations and asking you to determine which direction the rxn will proceed, and then possibly using those concentrations to do an ICE table.
So for your reaction, you can find the formula for your Ka. Once you set up your ice table with your initial values you can grab those values and plug them into your equation for Ka (concentration of products over concentration of reactants). Since your Q value tells you the initial (or current) state of the system, you plug in the values from the first row of your ice table into your equation.
You use ICE tables to find the concentrations at equilibrium because the whole point of an ICE table is to show the initial concentration, the change in concentration, and the equilibrium concentration. To find Q you just find the ratio of products to reactants at any point during the reaction.
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