He committed zina during the day in Ramadan, and is afraid that if he fasts two consecutive months he will be exposed to shame, so can he feed sixty poor persons?

Firstly:

What the person who is faced with this problem must do is three things:

1. He must repent to Allah, may He be exalted, from these two grave major sins, namely the shameful deed and deliberately breaking the fast during the day in Ramadan. Whoever repents, Allah will turn to him in mercy and change his bad deeds into good deeds, as He, may He be exalted, says (interpretation of the meaning):
2. And those who do not invoke with Allah another deity or kill the soul which Allah has forbidden [to be killed], except by right, and do not commit unlawful sexual intercourse. And whoever should do that will meet a penalty.

Multiplied for him is the punishment on the Day of Resurrection, and he will abide therein humiliated –

Except for those who repent, believe and do righteous work. For them Allah will replace their evil deeds with good. And ever is Allah Forgiving and Merciful”

[al-Furqaan 25:68-70].

1. He must make up the day on which he broke the fast.
2. He must offer expiation, which is to free a slave. If that is not possible, then he must fast for two consecutive months. If he cannot do that, then he must feed sixty poor persons, because of the report narrated by al-Bukhaari (1936) from Abu Hurayrah (may Allah be pleased with him) who said: Whilst we were sitting with the Prophet (peace and blessings of Allah be upon him), a man came to him and said: O Messenger of Allah, I am doomed! He said: “What is the matter with you?” He said: I had intercourse with my wife whilst I was fasting. The Messenger of Allah (peace and blessings of Allah be upon him) asked: “Can you afford to free a slave?” He said: No. He said: “Are you able to fast for two consecutive months?” He said: No. He said: “Can you afford to feed sixty poor persons?” he said: No. The Prophet (blessings and peace of Allah be upon him) paused for a while, and whilst we were like that, a large basket of dates was brought to the Prophet (blessings and peace of Allah be upon him). He said: “Where is the one who was asking?” The man said: I am here. He said: “Take it and give it in charity.” The man said: To someone poorer than me, O Messenger of Allah? By Allah, there is no family between the two lava fields [i.e., in Madinah] who is poorer than my family. The Prophet (blessings and peace of Allah be upon him) smiled so broadly that his eye-teeth became visible, then he said: “Feed your family with it.”

Ibn Qudaamah (may Allah have mercy on him) said: The expiation (kafaarah) becomes a binding obligation upon one who has intercourse in the front or back passage in Ramadan deliberately, whether he ejaculates or not, according to the view of the majority of scholars.

End quote from al-Mughni (3/60)

This expiation is mentioned in order of preference, according to the majority of fuqaha’, so it is not permissible to feed poor persons for one who is able to fast.

Secondly:

If you are able to fast, then feeding poor persons will not be acceptable in your case, and what you fear of people around you finding out is no excuse. You can vow to offer this expiatory fast, and then tell anybody who asks you that you vowed to fast for two consecutive months. In this manner you can avoid causing yourself any embarrassment.

Thirdly:

The fact that you were unaware at that age that this expiation is obligatory does not mean you are excused. If someone knows of the prohibition, but is unaware of its consequences, he is not excused.

 And Allah knows best.