Here is a sample of such quotes directly from the Quran:
"Believers, if you yield to the infidels they will drag you back to unbelief and you will return headlong to perdition. . . .We will put terror into the hearts of the unbelievers. . . . The Fire shall be their home" (3:149-51)
"Believers, do not follow the example of the infidels, who say of their brothers when they meet death abroad or in battle: ‘Had they stayed with us they would not have died, nor would they have been killed.' God will cause them to regret their words. . . . If you should die or be slain in the cause of God, God's forgiveness and His mercy would surely be better than all the riches they amass" (3:156).
"God will not forgive those who serve other gods besides Him; but He will forgive whom He will for other sins. He that serves other gods besides God is guilty of a heinous sin. . . . Consider those to whom a portion of the Scriptures was given. They believe in idols and false gods and say of the infidels: ‘These are better guided than the believers'" (4:50–51).
Yes there are also some quotes in the Quran that contradict these quotes (like all religious texts, it's highly contradictory), and some people may be animated by charismatic individuals who distort the text, but Muslims who commit violence in the name of Allah need not believe anything more than the specific claims that the Quran makes to rationalize their violent actions.
The solution to this problem is like asking, "What's the solution to someone who still believes in Santa Claus at the age of 30?"
If you encountered such a person, would you let them continue to believe in Santa Claus or would you empathetically attempt to communicate to that person that they need to deeply examine their beliefs and ensure they're based in reality, based on evidence that can be readily observed.
But when most atheists and agnostics encounter a Muslim (or a Christian, or a Jew, or any religious person), they simply roll their eyes and shrug it off because they don't "want to get into it with them."
I'm not saying you have to go around trying to prove everybody wrong all the time, that's a recipe for ensuring everyone hates you. But religious beliefs are dangerous and they are the direct cause of violence -- not just Muslims of course, the Crusades are an obvious example of Christianity being the direct cause of violence.
The solution is to be empathetic, but not apologetic about the fact that their religious beliefs are not rooted in reality, and all beliefs should be.
TL;DR -- When you encounter a religious person, be empathetic, but not apologetic when communicating that their beliefs are based on a made up story rather than rooted in reality (as all beliefs should be).