The general theory tells us that forms of a reductive group $G$ are classified by $H^1$ of the automorphism group, which is a semidirect product (root datum automorphisms) x (inner automorphisms). For $SL_n / k$ the root datum has no automorphisms, so the classifying object is $H^1(k, PGL_n(\overline{k})) \cong H^2(k, \overline{k}{}^\times)[n] = Br(k)[n]$. Hence, by standard results on the Brauer group, a form of $SL_2$ over a number field is uniquely determined by the set of places at which it's locally non-split, and the only constraint on this set is that it must have even size.
Now, let's look at unitary groups $U(V)$ over local fields. If $k$ is local and $K$ is a degree 2 étale ring extension of $k$, then $K$ is either $k \oplus k$ or a field extension. In the former case, any Hermitian space wrt $K/k$ is obviously split. In the latter case, the isomorphism class of $V$ is uniquely determined by the discriminant $disc(V) \in k^\times / Nm(K^\times)$. For your example, with matrix $diag(1, \lambda)$, the discriminant is $-\lambda$. When $V$ is split, then one computes that $SU(V) \cong SL(2)$, as in the familiar argument for $k = \mathbf{R}$ (of course $U(V)$ is not $GL(2)$ unless $K = k \oplus k$!)
Conversely: it's non-obvious, but true, that if $V$ is non-split then $SU(V)$ is also non-split. If it were split, then the Borel subgroup of $G$ would have to have a fixed point in $\mathbf{P}(V)$ by the Borel-Morozov theorem; and the restriction of the Hermitian form to this line has to be zero, hence $V$ is split.
Going back to the global case, if $K = k(\sqrt{\alpha})$, and $\beta = -\lambda$, we have shown that $SU(F_{K,\lambda})$ is split locally at $v$ if and only if $(\alpha, \beta)_{k_v} = 1$, where $(-, -)_{k_v}$ is the Hilbert symbol. In other words, $SU(F_{K,\lambda})$ is split locally at $v$ if and only if the quaternion algebra $D = (\alpha, \beta)_k$ splits there. So $SU(F_{K,\lambda})$ is isomorphic to $(D^\times)^{Nm = 1}$.
Since every quaternion algebra is $(\alpha,\beta)_{k}$ for some $\alpha$ and $\beta$, we conclude the answer to the question is "Yes".
[PS: With a little more work, one should be able to write down explicitly an isomorphism from $SU(F_{k(\sqrt{\alpha}),\lambda})$ to the norm 1 elements of $D^\times$ where $D = (\alpha, -\lambda)_k$, using the fact that $D$ is naturally a 2-dim'l vector space over $k(\sqrt{\alpha})$. But it's a Sunday morning so I'm going to be lazy and not do the exercise.]