How many times does the Bible state that Jesus of Nazareth, whether pre-incarnate or incarnate, was "begotten"?
Once - in two different ways depending on our definition of 'begotten'
There seems to be some confusion on the subject because "only begotten" a theological term does not mean "begotten" a biblical term.
But to answer your question, if thinking 'begotten' as in 'only begotten' it gains prominence in Council of Nicaea in A.D. 325 representing Christ’s single eternal generation:
Not to say that the theological term is not very biblical. When the word begotten is used with “son” it does mean God’s only Son, indicating God’s absolute delight towards him and his absolute uniqueness (Jn 3:16, 18; 1 Jn 4:9). The statement at the baptism and transfiguration of Jesus in the Synoptics 'This is my beloved son' also carries the same sense of God’s absolute delight towards him and his absolute uniqueness, but not principally a literal/spiritual 'birth'. Not 'principally' a 'birth' but by implication as Jesus is unique since he is also eternally begotten. If that does not make him unique I do not know what can. In fact the very notion of a son, and all its uses in scripture could be argued as having its ultimate meaning in dimly reflecting the glory of the Son. The concept of sonship finds its ultimate meaning in Christ.
To confuse matters more, in Hebrews Chapter 1:5 it reads ‘For to which of the angels did God ever say, “You are my Son, today I have begotten you”? Or again, “I will be to him a father, and he shall be to me a son”? this seems to imply a 'birth' or sorts, but not really. From this type of reference we are naturally tempted to mix the theological term (i.e. eternal generation) with the biblical prophecy first fulfilled in David (as a type) and in Christ as Messiah. Therefore, although “only begotten” is a theological term referring to a single eternal event, ‘today I have begotten you’ means something totally different in terms of a true generation, or 'birth'. This really refers to his 'birth' from the dead, or the first born.
John Owen explains the different views about 'begotten' very well here, as well as the correct view to take:
Conclusion: The biblical term “begotten” refers to the resurrection of Christ, which happened once. The theological term “only begotten” refers to the eternal generation of the Son, by an inconceivable communication of the essence and substance of the Godhead by the Father --- only happened in a continuous forever. God’s absolute delight towards Christ and his absolute uniqueness implied by the words 'only Son' brings the single eternal generation and the single generation from the dead together under the Sonship theme. Both his eternal generation and his resurrection make him absolutely unique and proven fully approved of by God. This is partly why Jesus always used the phrase 'the Son' as implying being one and only person generated from yet equal to his Father, for example, as in John 5:16-29.