If Q(e_1) = 1 and Q(e_2) = -1 (meaning that Q(e_1, e_1) = 1 and Q(e_2, e_2) = -1), then
Q(e_1+e_2)
= Q(e_1+e_2, e_1+e_2)
= Q(e_1, e_1) + 2Q(e_1, e_2) + Q(e_2, e_2)
= 1 + 2Q(e_1, e_2) - 1
= 2Q(e_1, e_2)
In your post above, you said that e_1 and e_2 form a basis. If this implies that Q(e_1, e_2) = 0, then, continuing from the above, we see that
Q(e_1+e_2) = 0.