“Calcibus” is the ablative or dative plural of calx, which means “limestone” (from which we get the term “calcium”) or, alternatively, “heel.”
I believe the translator mistook “calcibus” for “calicibus.”
If we take the “limestone” meaning, based on the context, the phrase almost certainly refers to the fact that many prophets were stoned to death because what they preached was uncomfortable. (See, e.g., Mt. 23:37.)
ut passiones martyrum imitetur, non eos calcibus persequamur!
would be rendered
In order that the sufferings of the martyrs may be imitated, let us not persecute them with stones.
If we take calx to mean heel, we would have:
In order that the sufferings of the martyrs may be imitated, let us not persecute them with [our] heels [i.e., let us not tread them underfoot].
Although the translation is uncertain, it certainly does not involve cups (calices, with an extra I).
Some other translations
A survey of translations favors the “heel” interpretation:
The English translation by Boniface Ramsey reads
And would that it might also dwell in our hearts such that we may imitate the martyrs’ sufferings and not trample them underfoot!
Likewise, the Italian translation on augustinus.it has
Potesse quel nome stabilirsi anche nel cuore, tanto da farci imitare le sofferenze dei martiri e non metterle invece sotto i piedi.
May that name be established also in the heart, so as to imitate the sufferings of the martyrs and not to tread them, instead, underfoot (a rather loose, but valid, translation, as can be seen).
The Spanish translation on the same site makes the same choice:
Y ¡ojalá habite también en el corazón de tal modo que, en vez de ponerlos bajo nuestros pies, imitemos sus pasiones!
May [the word martyr] live in your heart un such a way that, instead of putting them under our feet, we may imitate their sufferings!