We always thank God, the Father of our Lord Jesus Christ...

He has delivered us from the domain of darkness and transferred us to the kingdom of His beloved Son, in whom we have redemption, the forgiveness of sins. -- Colossians 1:3,13-14 (ESV)

Is "in whom we have redemption, the forgiveness of sins" referring to "God, the Father" or "His beloved Son"?

Colossians 1:3-12 is all about the Father, so it is likely that according to Paul, the Father is the Redeemer and forgiver of sins, however if you read the NKJV version, the words "through his blood" is added. Since we know the Father is not a human with blood, this disqualifies the Father, and we can only imply that Paul meant Jesus:

He has delivered us from the power of darkness and conveyed us into the kingdom of the Son of His love, in whom we have redemption through his blood, the forgiveness of sins. Colossians 1:13-14 (NKJV)

The next verse continues:

He is the image of the invisible God, the firstborn over all creation. -- Colossians 1:15 (NKJV)

The Father is not an "image of God" and neither was He "born", so Paul most likely refers to Jesus in this case.

The Father forgives sin

• The Father forgave Joseph's brothers' sin (Genesis 50:17)
• The Father forgave the Pharaoh's sin (Exodus 10:17)
• The Father forgave Israel's sin (Exodus 32:32; 1 Kings 8:30,34,36,49,50; 2 Chronicles 6:21,25,27,30,39; 7:13-14; Daniel 9:19)
• The Father forgave David's sin (Psalm 25:18; 86:5)

Jesus teaches that we should ask the Father to "forgive us our debts" as we too need to "forgive our debtors":

And forgive us our debts, as we forgive our debtors. -- Matthew 6:12 (NKJV)

Jesus forgives sin

And behold, some people brought to him a paralytic, lying on a bed.

And when Jesus saw their faith, he said to the paralytic, “Take heart, my son; your sins are forgiven.”
And behold, some of the scribes said to themselves, “This man is blaspheming.”
But Jesus, knowing their thoughts, said, “Why do you think evil in your hearts? For which is easier, to say, ‘Your sins are forgiven,’ or to say, ‘Rise and walk’? But that you may know that the Son of Man has authority on earth to forgive sins”
He then said to the paralytic: “Rise, pick up your bed and go home.”

And he rose and went home. When the crowds saw it, they were afraid, and they glorified God, Who had given such authority to men. -- Matthew 9:2-8 (ESV)

Note that Jesus forgave sin on earth with the authority of God.

Humans forgive sin

However, in each case the person forgave sin against himself. There are no examples of any Christian priests forgiving sin against God, except "the Son of Man" who received this authority from God.

Joseph's brothers asking forgiveness from Joseph

So they sent a message to Joseph, saying, “Your father gave this command before he died: ‘Say to Joseph, “Please forgive the transgression of your brothers and their sin, because they did evil to you.”’ And now, please forgive the transgression of the servants of the God of your father.”

Joseph wept when they spoke to him.

-- Genesis 50:16-17 (ESV)

Pharaoh asking forgiveness from Moses and Aaron

Then Pharaoh hastily called Moses and Aaron and said, “I have sinned against the LORD your God, and against you. Now therefore, forgive my sin, please, only this once, and plead with the LORD your God only to remove this death from me.”

So he went out from Pharaoh and pleaded with the LORD. And the LORD turned the wind into a very strong west wind, which lifted the locusts and drove them into the Red Sea. Not a single locust was left in all the country of Egypt.

-- Exodus 10:16-19 (ESV)

Abigail asking forgiveness from David

When Abigail saw David, she hurried and got down from the donkey and fell before David on her face and bowed to the ground. She fell at his feet and said, “On me alone, my lord, be the guilt. Please let your servant speak in your ears, and hear the words of your servant. Let not my lord regard this worthless fellow, Nabal, for as his name is, so is he. Nabal is his name, and folly is with him. But I your servant did not see the young men of my lord, whom you sent. Now then, my lord, as the LORD lives, and as your soul lives, because the LORD has restrained you from bloodguilt and from saving with your own hand, now then let your enemies and those who seek to do evil to my lord be as Nabal. And now let this present that your servant has brought to my lord be given to the young men who follow my lord. Please forgive the trespass of your servant. For the LORD will certainly make my lord a sure house, because my lord is fighting the battles of the LORD, and evil shall not be found in you so long as you live.” -- 1 Samuel 25:23-28 (ESV)

Jesus command forgiveness

For if you forgive men their trespasses, your heavenly Father will also forgive you. But if you do not forgive men their trespasses, neither will your Father forgive your trespasses. -- Matthew 6:14-15 (NKJV)

Jesus also told a parable:

Then Peter came to him and said, “Lord, how often shall my brother sin against me, and I forgive him? Up to seven times?”
Jesus said to him, “I do not say to you, up to seven times, but up to seventy times seven. Therefore the kingdom of heaven is like a certain king who wanted to settle accounts with his servants. And when he had begun to settle accounts, one was brought to him who owed him ten thousand talents. But as he was not able to pay, his master commanded that he be sold, with his wife and children and all that he had, and that payment be made. The servant therefore fell down before him, saying, ‘Master, have patience with me, and I will pay you all.’ Then the master of that servant was moved with compassion, released him, and forgave him the debt. But that servant went out and found one of his fellow servants who owed him a hundred denarii; and he laid hands on him and took him by the throat, saying, ‘Pay me what you owe!’ So his fellow servant fell down at his feet and begged him, saying, ‘Have patience with me, and I will pay you all.’ And he would not, but went and threw him into prison till he should pay the debt. So when his fellow servants saw what had been done, they were very grieved, and came and told their master all that had been done. Then his master, after he had called him, said to him, ‘You wicked servant! I forgave you all that debt because you begged me. Should you not also have had compassion on your fellow servant, just as I had pity on you?’ And his master was angry, and delivered him to the torturers until he should pay all that was due to him. So My heavenly Father also will do to you if each of you, from his heart, does not forgive his brother his trespasses.” -- Matthew 18:21-35 (NKJV)

"Pay attention to yourselves! If your brother sins, rebuke him, and if he repents, forgive him, and if he sins against you seven times in the day, and turns to you seven times, saying, ‘I repent,’ you must forgive him." Luke 17:3-4 (ESV)

Paul also requested forgiveness:

Put on then, as God's chosen ones, holy and beloved, compassionate hearts, kindness, humility, meekness, and patience, bearing with one another and, if one has a complaint against another, forgiving each other; as the Lord has forgiven you, so you also must forgive. -- Colossians 3:12-13 (ESV)