Mazur's theorem is not needed here, and besides, it applies only to elliptic curves themselves defined over $\mathbb{Q}$ rather than the said compositum $K^{(2)}$ of all quadratic extensions of $\mathbb{Q}$ (which is how I understand the question).
A simple proof is available by purely local methods. First, if $F/\mathbb{Q}_p$ is a finite extension, then any abelian variety $A/F$ has $|A(F)_{\mathrm{tors}}| < \infty$. This follows from the theory of the formal group; for the elliptic curve case, see Chapter IV, Theorem 6.4 in Silverman's book (supplemented by Ch. VII Ex. 7.6 as noted by René).
Given this, fix any prime. Since $\mathbb{Q}_p$ has just finitely many quadratic extensions, it is possible to embed $K^{(2)}$ in a finite extension $F/\mathbb{Q}_p$. Then the stated finiteness follows from the above quoted result.