First of all I have to point out that the word “quod” in book II, line 141 of Vergil's Aeneid is not an adverbial accusative, but simply a causal conjunction introducing the causal clause with the verb “oro”, as you can read in the literal translation at the foot of my answer.
Your notes describe “quod” as an 'adverbial accusative' because the causal conjunction “quod”, (meaning “because”/” for”/ “ since”) was in origin a relative pronoun used adverbially in the accusative neuter , though its use as a causal particle is an early special development.
As for the Latin 'adverbial accusative' , it is an idiomatic use in a few adverbial phrases such as “id temporis”( at that time), “maximam partem”( for the most part), “quod nisi” ( if not), etc.
(See Allen and Greenough's New Latin Grammar for Schools and Colleges
J. B. Greenough, G. L. Kittredge, A. A. Howard, Benj. L. D'Ooge, Ed., §397 at http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0001%3Apart%3D2%3Asection%3D9%3Asubsection%3D13%3Asmythp%3D397 ).
So, here’s the literal translation of “Quod te per superos et conscia numina veri,/per si qua est quae restet adhuc mortalibus usquam/ intemerata fides, oro, miserere laborum/ tantorum, miserere animi non digna ferentis”:
“Because ( quod) I beg (oro) you (te) in gods above name (per superos) and the deities (numina, accusative neuter plural depending on “per) [who are] conscious (conscia, acc neuter plural agreeing with “numina”) of the truth (veri, genitive sing), in name of (per) a pure (intemeratam, which is implied, however, just like the following “fidem”, both depending on the preposition “per”) faith (fidem, implied), if (si) some (qua/aliqua) pure faith (intemerata fides, nominative) exists (est) which (quae) still (adhuc) remains (restet, present subjunctive suggesting possibility) anywhere (usquam) among the mortals (mortalibus, dative depending on “restet”), have pity (miserēre, 2nd sing present imperative of the deponent verb “misereor”) on so great sufferings (tantorum laborum, genitive plural depending on “miserēre”), have pity (miserēre) on a soul (animi, genitive singular depending on “miserēre”) which suffers (ferentis, present participle in the genitive singular agreeing with “animi”) not deserved things (not digna, acc neuter plural)”, i.e.:
“So, I beg you in gods name as well as in the name of those deities who know the truth, and also in the name of a pure faith, if some pure faith exists which still remains anywhere among the mortals, have pity on my so great sufferings, have pity on my heart which suffers what I really do not deserve”.
Please note that:
in the literal translation I’ve used round brackets to indicate Latin terms and square brackets to indicate implied terms.
in “per si qua est quae restet adhuc mortalibus usquam /intemerata fides” (line 142) the preposition “per” implies the accusative “intemeratam fidem” which is understood, because “per” is used with ellipsis of the object in order to avoid any repetition.
Hope all is clear enough.