Let $\mathcal T$ be a triangulated category and $\mathcal C$ a thick triangulated subcategory. We consider the Verdier quotient $\mathcal T/\mathcal C$.
Is there a bijective correspondence between thick triangulated subcategories of the quotient $\mathcal T/\mathcal C$ and thick triangulated subcategories of $\mathcal T$ containing $\mathcal C$?
It "feels" true, but I have learned never to take chances with technicalities of triangulated categories.
Yes, this is true and works as you expect. See Proposition 2.3.1 (pages 125-127) of Verdier's thesis:
Since it is in French, let me just remark that "sous-catégorie triangulée pleine" means "full triangulated subcategory", "sous-catégorie triangulée strictement pleine" means "strictly full triangulated subcategory" a.k.a "full triangulated subcategory closed under isomorphism", and Verdier uses "saturée" to mean "thick" (closed under direct summmands).
Let $\mathcal D$ be a triangulated category, $\mathcal B\subset \mathcal D$ a thick triangulated subcategory, and let $Q:\mathcal D \rightarrow \mathcal D / \mathcal B$ denote the Verdier quotient. That proposition establishes a bijection between the (strictly full) triangulated subcategories of $\mathcal D$ which contain $\mathcal B$ and the (strictly full) triangulated subcategories of $\mathcal D/\mathcal B$. It restricts to a bijection between the (strictly full) thick triangulated subcategories of $\mathcal D$ which contain $\mathcal B$ and the (strictly full) thick triangulated subcategories of $\mathcal D/\mathcal B$.
