zeugen
to father
For a man to biologically produce a child.
Er hat mit seiner Frau zwei Kinder gezeugt.
He fathered two children with his wife.

to father / to testify
Loading your saved state.
to father
For a man to biologically produce a child.
Er hat mit seiner Frau zwei Kinder gezeugt.
He fathered two children with his wife.

'Zeugen' (to father / to testify) is related to 'der Zeuge' (witness) and 'das Zeugnis' (testimony/certificate). The shared idea is bringing something forth or proving something—whether a child, a statement, or an achievement.
Seine guten Noten sind ein Zeugnis für seinen Fleiß.
His good grades are testimony to his diligence.
In the meaning 'ein Beweis sein für' (to be evidence of), 'zeugen' (to testify to) is always used with the preposition 'von' and the dative. The structure is: 'Etwas zeugt von etwas' (Something testifies to something).
Ihre schnelle Reaktion zeugt von großer Professionalität.
Her quick reaction testifies to great professionalism.
The meaning 'eine Aussage machen' (to make a statement) is found almost exclusively in a legal context. Typical formulations are 'vor Gericht zeugen' (to testify in court) or 'gegen jemanden zeugen' (to testify against someone).
Er musste vor Gericht gegen seinen ehemaligen Freund zeugen.
He had to testify in court against his former friend.
In the meaning 'ein Kind hervorbringen' (to produce a child), 'zeugen' (to father) is a rather formal or biological term. It is often used in written or formal contexts.
Laut Testament hat der Verstorbene drei Kinder gezeugt.
According to the will, the deceased fathered three children.
to show
'zeigen' and 'zeugen von' can both mean that something is a sign or proof of something. 'zeigen' is much more common and simpler in everyday language: a behavior can show respect, and a result can show a problem. 'zeugen von' sounds more elevated and is almost always used with 'von'. It fits well when something points to a characteristic, attitude, or quality. Learners may confuse the words because the meaning is similar. Important: in this meaning, you say 'zeugen von etwas' (to testify to something), not simply 'zeugen'.

Die vielen Bücher im Zimmer zeugen von seinem großen Interesse an Geschichte.
The many books in the room testify to his great interest in history.
Ihre schnelle Hilfe zeigt, dass sie sehr verantwortungsvoll ist.
Her quick help shows that she is very responsible.