To get its character table first show that Q has 5 conjugacy classes.
Then find all its 1 dimensional reps as given below..
First prove that . We know that all the one dimensional reps of Q can be obtained from the one dimensional reps of Q/[Q,Q] (which is isomorphic to Klein 4 group). I assume you know the one dimensional reps of Klein 4 group. (hint: Just try all feasible combination of 1 and -1 in the table).
Then use sum of (character of identity)^2 = #Q = 8 to figure out the only non 1 dimensional irrep's dimension is 2.
Now use orthogonality relations to fill the table. You will observe non 1 dimensional rep's character has a peculiar nature which will help you figure out the irrep