He said that he committed zina with a woman but he was lying

We ask Allaah to forgive your sin and to
conceal your fault, and to help you avoid temptations both obvious and
hidden. With regard to your question, it can be answered in the following
points: 

1 – It seems from your question that the one
whom you accused of committing forbidden actions is a specific woman whom
you and your cousin know. If it is a specific, known, woman, then this is
slander by you against her, which is a major sin and an abhorrent forbidden
action, because it is a slander against her honour and a transgression of
the limits set by Allaah. The Prophet (peace and blessings of Allaah be
upon him) said, as was narrated by Abu Hurayrah (may Allaah be pleased with
him): “Avoid the seven sins that doom one to Hell.” They said: “O Messenger
of Allaah, what are they?” He said: “Associating others with Allaah (shirk),
witchcraft, killing a soul whom Allaah has forbidden us to kill, except when
it is lawful, consuming riba, consuming the orphan’s wealth, running away
from the battlefield, and slandering chaste believing women who never even
think of anything touching their chastity (cf. al-Noor 24:23).” Narrated by
al-Bukhaari, no. 2615; Muslim, 89. 

As well as slandering that woman, you are
also saying that you yourself committed zina, even if you are lying. 

But if it is not a specific women, for
example you said, “I did such and such with a woman,” and did not refer to a
specific woman, then this is not regarded as slandering anyone, but you are
still saying that you committed zina for which a hadd punishment is
deserved, so you have to clear your own name.  

It says in Badaa’i’ al-Sanaa’i’: If a
person says, “I committed zina with a woman, but I do not know who she
was,”, his confession is valid and he should be punished. Badaa’i’
al-Sanaa’i’, 7/51 

2 – If it is a specific woman, but you have
repented and regret what you did, part of your repentance is to tell others
that you were lying, so you have to tell your cousin that you were not
telling the truth, because you have to clear the name of that chaste woman.
It is not permissible for you to give precedence to the friendship with your
cousin by slandering the honour of a Muslim woman. 

Ibn Qudaamah said in al-Mughni: 

The apparent meaning of Ahmad’s words
is that the repentance of the slanderer is to state that he was lying. So he
should say, “I was lying when I said that.” This is the view of al-Shaafa’i.
Ibn ‘Abd al-Barr said: Those who said that also include Sa’eed ibn
al-Musayyib, ‘Ata’, Tawoos, al-Shu’bi, Ishaaq, Abu ‘Ubayd and Abu Thawr.

‘Abd al-Razzaaq narrated in al-Musannaf
(5/77) that Tawoos said: His repentance (i.e., the one who accused someone
of zina) is to say that he was lying. 

3 – You should note that the Prophet
(peace and blessings of Allaah be upon him) said: “Whoever seeks the
pleasure of Allaah by angering people, Allaah will suffice him against
people, but whoever seeks to please people by angering Allaah, Allaah will
leave him to the people.” Narrated by al-Tirmidhi, no. 2414; classed as
saheeh by al-Albaani.

This hadeeth indicates that what you fear of
a bad relationship between you and your cousin is an insignificant matter.
Whoever obeys Allaah and seeks to please Him, Allaah will suffice him
against people. So you should care more about your relationship with your
Lord than your relationship with your cousin. Remember that this is one of
the trials with which Allaah tests you, so do what is right and put your
trust in Allaah, and do not fear the blame of anyone. Pray a great deal for
forgiveness, repent and make du’aa’ to Allaah, asking him to make you
steadfast in speaking the truth, and to reconcile you and your cousin, for
the hearts of men are between two of the fingers of the Most Merciful, and
He turns them however He wills. And Allaah knows best.