No, your cousin is not right -- and in fact he is deceiving himself by thinking he can live any way he wants to and still get to heaven. One of the devil's oldest tricks is to deceive us into thinking God doesn't care how we live.
The saying you quote isn't found in the Bible -- although it contains much truth. God does love us -- even when we don't love Him in return. But He doesn't love our sin -- and one reason is because He knows it will ultimately destroy us. Just as earthly parents continue loving a wayward child in spite of their rebellion, so our Heavenly Father continues loving, in spite of our rebellion.
God loves us so much, in fact, that He sent His Son into the world to make it possible for us to be forgiven. Would Jesus Christ have gone to the cross if sin didn't matter, and God was going to take us to heaven anyway? No, of course not. The Bible says, "God demonstrates his own love for us in this: While we were still sinners, Christ died for us" (Romans 5:8).
Pray for your cousin, and ask God to help you be both loving and honest with him. He needs Christ, and hopefully once he understands what it cost Christ to bring us our salvation, he'll want to repent of his sins and open his heart to Christ.