He came home from travelling and was not fasting, and he had intercourse with his wife during the day in Ramadan; do they have to offer expiation?

Praise be to Allah.

If a traveller comes home not fasting, or a sick person recovers, or a menstruating woman becomes pure during the day in Ramadan, they do not have to refrain from eating and drinking, etc., according to the opinion of the majority of fuqaha’. See the answer to question no. 49008

But the one who is not fasting in such cases does not have the right to have intercourse with his wife who is fasting and was not travelling. If he does that and is aware of and remembers the ruling, then he is sinning by helping her and calling her to commit an act of disobedience. 

The case of the wife is subject to further discussion: 

-If at the time of intercourse she was excused because she was forced or forgot, or was unaware of the prohibition on intercourse during the day in Ramadan, then her fast is still valid, and she does not have to make it up or offer expiation according to the correct view.

-If she participated willingly, and was aware of and remembered the ruling, then she has sinned and invalidated her fast, and she has to offer expiation, according to the majority of the fuqaha’, because it is proven in as-Saheehayn that the Prophet (blessings and peace of Allah be upon him) instructed the man who had intercourse with his wife during the day in Ramadan to offer expiation. The basic principle is that rulings apply equally to both men and women, except in cases where the wise Lawgiver has clearly made an exception. Because she has transgressed against the Ramadan fast by having intercourse, then she must offer expiation like the man. The punishment has to do with intercourse, so it applies equally to both the man and the woman, as in the case of the hadd punishment for zina (fornication or adultery).

See also the answer to question no. 106532

And Allah knows best.