Learn Icelandic: Þau eiga gott samband. - They have a good relationship.

You certainly could say Þau eiga gott samband við hvort annað (They have a good relationship with each other). However, just like in English, the "with each other" is usually left out because the noun samband (relationship) combined with a plural subject already makes the reciprocal meaning obvious.

Why do we use eiga instead of hafa to mean "have" here?

In Icelandic, eiga (which literally means "to own") is the standard verb for personal relationships, friends, and family. You use it to say you "have" a brother, a partner, or in this case, a relationship (samband). The verb hafa is kept for more abstract concepts or temporary states.

Why does the sentence use the neuter pronoun þau for "They"?

In Icelandic, the neuter plural þau is always used when "they" refers to a mixed-gender group (like a man and a woman). If the relationship were between two men, you would use the masculine þeir, and for two women, the feminine þær.

What case is gott samband in?

It is in the accusative case because it is the direct object of the verb eiga. However, because samband is a neuter noun, its accusative form looks exactly the same as its basic dictionary (nominative) form. The adjective gott is in the neuter singular accusative to match.