The subjects of いう and 納得される are unspecified, but they are the same person. 職業が教師(だ)という is not "say teacher" but "say that my/his profession is a teacher".
職業が教師といえば納得される。
If he says his profession is a teacher, he will be understood.
If I say his profession is a teacher, I will be understood.
If one says his profession is a teacher, they will be understood.
Assuming the subject is the speaker's father:
如何にも職業が教師といえば納得される感じの清廉潔白な厳しさ
incorruptible strictness that is exactly like "if he says his profession is a teacher, he will be understood"
But you can rephrase this phrase without specifying who said "教師(だ)":
incorruptible strictness that indeed makes one agree when told his profession is a teacher
Note that 如何にも modifies 納得される感じ(だ).