There's no need to require irreducibility. If $A=E_1 \times E_2$ is a product of elliptic curves the conjecture is true, by modularity of elliptic curves combined with Yoshida's lifting from pairs of classical cusp forms to Siegel modular forms of genus two. If $K$ is a real quadratic field and $E/K$ is a modular elliptic curve, then Yoshida's conjecture is true for the surface $A=\mathrm{Res}_{K/\mathbb{Q}}(E)$. This follows from a theorem of Johnson-Leung and Roberts; see arxiv 1006.5105. Presumably any individual $A$ can be done "by hand" using Faltings-Serre plus some serious computational cleverness.
Sorry if you already know all these things. :)
What you are looking for is the correspondence between algebraic Hecke characters over a number field $F$ and compatible families of $l$-adic characters of the absolute Galois group of $F$. This is laid out beautifully in the first section of Laurent Fargues's notes here.
EDIT: In more detail, as Kevin notes in the comments above, an automorphic representation of $GL(1)$ over $F$ is nothing but a Hecke character; that is, a continuous character
$$\chi:F^\times\setminus\mathbb{A}_F^\times\to\mathbb{C}^\times$$
of the idele class group of $F$. You can associate $L$-functions to these things: they admit analytic continuation and satisfy a functional equation. This is the automorphic side of global Langlands for $GL(1)$.
How to go from here to the Galois side? Well, let's start with the local story. Fix some prime $v$ of $F$; then the automorphic side is concerned with characters
$$\chi_v:F_v^\times\to\mathbb{C}^\times$$
Local class field theory gives you the reciprocity isomorphism
$$rec_v:W_{F_v}\to F_v^\times,$$
where $W_{F_v}$ is the Weil group of $F_v$. Then $\chi_v\circ rec_v$ gives you a character of $W_{F_v}$. This is local Langlands for $GL(1)$. The matching up local $L$-functions and $\epsilon$-factors is basically tautological.
We return to our global Hecke character $\chi$. Recall that global class field theory can be interpreted as giving a map (the Artin reciprocity map)
$$Art_F:F^\times\setminus\mathbb{A}_F^\times\to Gal(F^{ab}/F),$$
where $F^{ab}$ is the maximal abelian extension of $F$. Local-global compatibility here means that, for each prime $v$ of $F$, the restriction $Art_F\vert_{F_v^\times}$ agrees with the inverse of the local reciprocity map $rec_v$.
Since $Art_F$ is not an isomorphism, we do not expect every Hecke character to be associated with a Galois representation. What is true is that $Art_F$ induces an isomorphism from the group of connected components of the idele class group to $Gal(F^{ab}/F)$. In particular, any Hecke character with finite image will factor through the reciprocity map, and so will give rise to a character of $Gal(F^{ab}/F)$. This is global Langlands for Dirichlet characters (or abelian Artin motives).
But we can say more, supposing that we have a certain algebraicity (or arithmeticity) condition on our Hecke character $\chi$ at infinity. The notes of Fargues referenced above have a precise definition of this condition; I believe the original idea is due to Weil. The basic idea is that the obstruction to $\chi$ factoring through the group of connected components of the idele class group (and hence through the abelianized Galois group) lies entirely at infinity. The algebraicity condition lets us "move" this persnickety infinite part over to the $l$-primary ideles (for some prime $l$), at the cost of replacing our field of coefficients $\mathbb{C}$ by some finite extension $E_\lambda$ of $\mathbb{Q}_l$. This produces a character
$$\chi_l:F^\times\setminus\mathbb{A}_F^\times\to E_\lambda^\times$$
that shares its local factors away from $l$ and $\infty$ with $\chi$, but now factors through $Art_F$. Varying over $l$ gives us a compatible family of $l$-adic characters associated with our automorphic representation $\chi$ of $GL(1)$. The $L$-functions match up since their local factors do.
Best Answer
The Taniyama conjecture says that the L-series of an elliptic curve over Q is automorphic (more specifically, arises from a modular form). Langlands conjectures that every L-series arising from algebraic geometry is automorphic (in the sense he defined).