Puisqu’il s’agit d’une locution latine, elle être utilisée telle quelle, et elle l’est de temps à autre.

Il n’y a cependant pas de « traduction standard » pour quid pro quo. Toute locution qui rend l’idée d’un échange quelconque passe. Une expression possible est donnant donnant.

Donnant donnant

En emploi de gérondif, loc. proverbiale. Pour signifier qu’on ne veut donner une chose qu’en recevant une autre chose.

C’est personnellement ce que j’aurais utilisé pour l’extrait du Silence des agneaux, mais Lecter parle plutôt d’échanges de bons procédés.

On peut également parler de contrepartie ou de compensation, tels que les mentionnent Termium et le Wiktionnaire.

I generally try to avoid Latin phrases, but when I encounter “quid pro quo” in English, I generally interpret it as capturing the notion of “the reciprocation/return of favors/actions” and my two preferred informal/familiar phrases for expressing this notion are:

“scratch my back, I’ll scratch yours” (always for favors) and
“tit for tat” (more neutral, sometimes for favors, but sometimes for
vengeful actions).

These two English phrases lead me to the respective notions of “renvoyer l’ascenseur” (always positive, as far as I know) and “rendre la pareille” (neutral) in French. Therefore, perhaps a French Hannibal could have said:

“[Principe de] renvoyer l’ascenseur”;
“[Principe de] la pareille
rendue/rendre la pareille”;

or even, going back to the notion of “reciprocation” for a more formal option:

“[L’idée/principe de] réciprocité” (instead of “quid pro quo.”)

(but see the clip provided in Kareen’s good answer as well as these two translations where Hannibal, apparently in two other scenes, uses “quid pro quo” in French.)

Inspired by un renvoi d’ascenseur I found useful to also cite the more familiar (but neutral) un prêté pour un rendu.

It would just post that in a comment, but I suddenly remembered how Coluche (a French humorist which died 30 years ago) had turned this formula, and I thought it would be interesting for our English readers to have a comprehension of this wordplay.

Rather than un prêté pour un rendu, Coluche said un prêté pour un vomi.
He played with the babyish sense of rendre for regurgitate or vomit, so giving a paradoxical (hence comic) tone to the formula, since the positive sense of rendu becames negative with vomi.