Who gets married at the beginning of "The Odyssey." book 4
I assume you are asking about the beginning of Book 4 and not about the beginning of the whole epic. No one is getting married at the beginning of the Odyssey as a whole.
At the beginning of Book 4, many people are gathered at the palace of King Menelaus. Both of his children, Megapenthes (a man) and Hermione are getting married (not to each other).
Their wedding feasts are an opportunity for the people present to reminisce about the Trojan War (Helen, wife of Menelaus, was the cause of the Trojan War and she talks about her experiences) and about Odysseus.
At the beginning of Book 4, Odysseus' son Telemachus and his companion Pisistratus (one of Nestor's sons), have arrived in Sparta. They hope to ask Menelaus, the king of Sparta, if he has any information about Odysseus' whereabouts. They happen to arrive when the Spartans are in the middle of celebrating the double weddings of Menelaus's two children--his son Megapenthes and his daughter Hermione. Megapenthes, who is Menelaus' son by a servant woman, is marrying Alector of Sparta's daughter. Hermione, his first and only child with the famous Helen, is marrying Neoptolemus, the son of Achilles. This union is a result of the events of the Trojan War. It is an interesting aside that the beautiful golden daughter of Helen of Troy (the instigator of the Trojan War) and Menelaus is going to marry the son of the most famous warrior of the Trojan War. Although the Odyssey does not mention this part, it is also an interesting fact that Achilles and Helen did not get along. He blamed her for starting a war that he did not want to fight in. The double weddings allow everyone to start telling stories of the Trojan War, which brings the narrative back to the figure of Odysseus, whose clever Trojan Horse trick was the winning stratagem of the war.