$\def\D{\mathcal{D}} \def\ind{\operatorname{Ind}} \def\K{\mathcal{K}} \def\A{\mathcal{A}}$Inside [GW, Remark F.168, p. 794], we find:

[Let $\K$ be a category and let $\K_S$ be its localization with respect to a right multiplicative system $S$.] If $\iota_\K:\K\to\ind(\K)$ is the fully faithful canonical functor and $Q:\K\to\K_S$ is the localization functor, one obtains a diagram of functors $$ \begin{matrix} \K&\xrightarrow{Q}&\K_S\\ \iota_\K&\searrow&\downarrow&\alpha\\ &&\ind(\K) \end{matrix} $$ which is not commutative. But there exists a natural morphism $$ \tag{F.42.1}\label{nat} \iota_\K\to\alpha\circ Q. $$ Let us suppose that $\K$ is a triangulated subcategory of $K(\A)$ for an abelian category $\A$ and that $\K_S$ is the localization by the system $S$ of quasi-isomorphisms in $\K$. Then $\ind(\K)$ is also triangulated, all functors above are triangulated and the morphism \eqref{nat} is a morphism of triangulated functors.

Here $\alpha:\K_S\to\ind(\K)$ is the canonical embedding of the localization of a category into its ind-completion, see e.g. [KS, Proposition 7.4.1]. Namely, $\alpha$ is the left Kan extension of $\iota_\K$ along $Q$ [KS, Proposition 7.3.3].

My questions are:

(Q1). Given a triangulated category $\K$, is there some canonical triangulated structure on its ind-completion $\ind(\K)$ turning $\K\to\ind(\K)$ into a triangulated functor?

(Q2). If the answer is to (Q1) is 'no' in general but 'yes' if $\K$ is a full subcategory of $K(\A)$, where $\A$ is abelian (as [GW] claims in the quote above), do you know any reference for this? And same for [GW]'s claims that $\iota_K$ and $\alpha$ are triangulated functors and \eqref{nat} is a morphism of triangulated functors. In [KS] there's nothing on these issues.

References

[GW]. Görtz, Wedhorn, Algebraic Geometry II

[KS]. Kashiwara, Schapira, Categories and Sheaves