You're asking if $Ext^1_G(\alpha,\beta)$ is one-dimensional. The short answer is no,
yet there are many cases where the answer is yes. Actually, the dimension of this group is not known in all cases, and when it is known, in general it is by a deep theorem,
not a trivial computation.

To be more precise, we have $Ext^1_G(\alpha,\beta) = Ext^1_G(1,\alpha^{-1} \beta) = H^1(G,\alpha^{-1} \beta)$ (continuous group cohomology). Setting $\chi=\alpha^{-1} \beta$
one is reduce to compute $H^1(G,\chi)$.  

The answer will depends mainly about the sign of $\chi$, that is $\chi(c)$ where $c$ is a complex conjugacy in $G$. Let us assume for simplicity  that there is no prime $p$
where you character $\chi$ restricted to $Gal(\overline{ \mathbb Q_p} /\mathbb Q_p)$ is equal
to the cyclotomic character. Also assume that $\chi$ is not trivial, because you have already found the right answer in that case (presumably using class field theory). 

> Then
the *expected* answer is $H^1(G,\chi)$ has dimension $1$ if $\chi$ is odd, $0$ if $\chi$ is even. 

When is it expected answer a theorem ? 

(a) For all $\chi$, when the prime number $l$ is regular in the sense of Kummer. This follows from Mazur-Wiles's proof of the main conjecture of Iwasawa, and the fact that the $l$-adic zeta function has no zero when $l$ is regular.

(a') for a given $l$, for all $\chi$ but a finite number (same argument, using the fact that the $l$-adic zeta function has finitely many zero in any case, being an Iwasawa function).

(b) When $\chi = \chi_{cycl}^n \epsilon$, where $n$ is an integer, $\epsilon$ a finite order character, excepted if $n$ is negative and $\chi$ even. This is due to a deep theorem of 
Soulé combined with some lemmas of Bloch-Kato.

As far as I know, these are the only known results. 

In the case we have left aside, the answer may be more complicated. For instance $H^1(G,\chi_{cycl})$ is of infinite dimension. To a more thorough discussion, please see my lecture notes on the Bloch-Kato conjecture for the Hawaii 2009's conference, available
on my web page.