True and good friendship must not be suspicious of anything.
It sounds unnatural although understandable.
Inserting また between 真の and 良き is very wordy. また means “or”, and it functions in this particular sentence as “the true, IN OTHER WORDS, good friendship”. Drop また from the Japanese sentence.
友情は～疑う is also grammatically incorrect in Japanese because 友情 is not people (i.e. living creatures) but the status of human relationship (i.e. concept). The act of 疑う should be taken by human beings.
～べきではない doesn’t capture the nuance of “must”. I believe the English sentence with “must” means that “there is no doubt that…” or “it’s certain that…”. But ～べきではない in the Japanese sentence means the obligation of behaving in good faith.
My alternative translation is:
I intentionally swapped the subject and the complement because it sounds more natural in Japanese. I also translated “not ~ anything” as いかなる状況でも (lit. under any circumstance).