Yes, incest (sex between close relatives) can be forgiven. All sins (except blasphemy against the Holy Spirit: Mark 3:29) can be forgiven; as Jesus said in Mark 3:28:
Truly, I say to you, all sins will be forgiven the children of man…
Examples of terrible sins being forgiven
King David was an adulterer and, according to the Law of Moses, he should have been killed for committing adultery (Lev.20:10) — but God forgave him (2Sam.12:13).
Manasseh was possibly the most evil Jewish king of all (2 Kings 21:2-6,16), but God forgave him (2 Chronicles 33:11-13).
To be forgiven, repent
With both David and Manasseh, God forgave them because they repented (David: 2Sam.12:13 & Ps. 51; Manasseh: 2 Chronicles 33:12-13). If we want to be forgiven then we need to repent as well:*
…”Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins… (Acts 2:38)
Repentance is when we are sorry about our sins and we want to stop doing them, when we want to do the right things instead of sinning:
Bear fruit in keeping with repentance.
New Living Translation
Prove by the way you live that you have really turned from your sins and turned to God.
So, if we repent, we can be forgiven.
If we confess our sins, he is faithful and just and will forgive us our sins and purify us from all unrighteousness. (1 John 1:9 NIV)
* Notice that the following quote — Acts 2:38 — says we also need to be baptised for our sins to be forgiven. This is something that a Christian only needs to do once, at the start of their life as a Christian. For this answer I am presuming that the person has already been baptised.