The conundrum of Isa's Father

Question

Surah 19:88 mentions, "And they say, 'The Most Merciful has begotten a son.'"

Does this mean that God was His Father?

Answer 1

No. It means that that's what "they" say, "they" being the Christians in this case. In context, the entire rest of the Surah is spent describing in no uncertain terms how wrong "they" are with that allegation.

Answer 2

No, one of the fundamental beliefs of Islam about Allah is that:

He neither begets nor is begotten

^{— Quran 112:3}

Regarding your query on 19:88, as other answers have mentioned, and is apparent from the wording of the verse itself: the Quran is narrating to us what some people believe, and is negating this belief in the subsequent verses:

And it is not appropriate for the Most Merciful that He should take a son.

There is no one in the heavens and earth but that he comes to the Most Merciful as a servant.

^{— Quran 19:92-93}

The same negation is present elsewhere, for example:

... and the Christians say, "The Messiah is the son of Allah ." That is their statement from their mouths; they imitate the saying of those who disbelieved before them. May Allah destroy them; how are they deluded?

^{— Quran 9:30}

The Messiah, Jesus, the son of Mary, was but a messenger of Allah and His word which He directed to Mary and a soul [created at a command] from Him. So believe in Allah and His messengers. And do not say, "Three"; desist - it is better for you. Indeed, Allah is but one God. Exalted is He above having a son.

^{— Quran 4:171}

The messiah, son of Mary, was not but a messenger; [other] messengers have passed on before him. And his mother was a supporter of truth. They both used to eat food. Look how We make clear to them the signs; then look how they are deluded.

^{— Quran 5:75}

Jesus was not but a servant upon whom We bestowed favor, and We made him an example for the Children of Israel.

^{— Quran 43:59}

Allah has not taken any son, nor has there ever been with Him any deity.

^{— Quran 23:91}

It is not [befitting] for Allah to take a son; exalted is He! When He decrees an affair, He only says to it, "Be," and it is.

^{— Quran 19:35}

As for Isa's father, he doesn't have any , his creation is similar to Adam who was created without a father or a mother:

Indeed, the example of Jesus to Allah is like that of Adam. He created Him from dust; then He said to him, "Be," and he was.

^{— Quran 3:59}

Answer 3

You quoted only verse 88, see the next 4 more, it will be clear

88: And they say, "The Most Merciful has taken [for Himself] a son."

And meaning of next 4 verses:

89: You have done an atrocious thing.

90: The heavens almost rupture therefrom and the earth splits open and the mountains collapse in devastation

91: That they attribute to the Most Merciful a son.

92: And it is not appropriate for the Most Merciful that He should take a son

Half knowledge is very dangerous!