Now you need to induce this one dimensional representation to two dimensional representation.
Use this induced representation formula, where is your one dimensional representation. Here, x_1, ..., x_t denotes the representatives for distinct left cosets of H in G. Thus choose x_1 = j and x_2 = 1.
Now you need to find the representation of each element in Q_8 using the above formula. Verify that it is an indeed representation.
You also need to show that the representation that you'll obtain is irreducible. If the is the character of your representation , then verify that in order to show it is irreducible.