The first questioning of the authority of God's word was by the serpent in the garden of Eden, and it began with "Yea, hath God said...?"
And so it continues.
Those who are blinded by Satan will always question the authority of God's word, just like their spiritual father did many, many years ago.
I've actually told some people who have questioned the authority of God's word just what I finished stating here. That's one approach, but it won't win you too many invitations to parties unless you're what's on the menu.
We could also, with Romans chapter 2 in mind, appeal to the God-given consciences of people even if they don't believe that the Bible is the word of God:
Romans chapter 2
[12] For as many as have sinned without law shall also perish without law: and as many as have sinned in the law shall be judged by the law;
[13] (For not the hearers of the law are just before God, but the doers of the law shall be justified.
[14] For when the Gentiles, which have not the law, do by nature the things contained in the law, these, having not the law, are a law unto themselves:
[15] Which shew the work of the law written in their hearts, their conscience also bearing witness, and their thoughts the mean while accusing or else excusing one another)
[16] In the day when God shall judge the secrets of men by Jesus Christ according to my gospel.
After explaining to people that God has given them a conscience, and that the word "conscience" literally means "with (con-) knowledge (science)", I have sometimes coupled this with the reminder that we're all going to die one day. I then explain to them that their own consciences have convicted them of sin, and that the wages of sin is death, and then attempt to bring the gospel into play. When reminded of the reality of their own mortality, some people will listen. Others, unfortunately, will not.
My normal approach, however, is to just preach the gospel to them anyway, whether or not they believe that the Bible is true. The word of God is "the sword of the Spirit" (Eph. 6:17), whether or not people believe it to be so, and it will still "pierce even to the dividing asunder of soul and spirit, and of the joints and marrow, and be a discerner of the thoughts and intents of the heart" (Heb. 4:12) even if people claim not to believe in it. I've known MANY unbelievers whom I've spoken the word of God to, and the "sword" that I stuck in them years ago is still stuck inside of them today. I've literally had people comment on things that I told them 20 years earlier as if I had just told them 5 minutes ago. God's word is effective...even in unbelievers.
If someone is a little more open to God's word, but is simply unsure of its authenticity, then I usually go the route of sharing Biblical prophecy and its fulfillment with them. Lately, that has been my primary method of trying to reach unbelievers because, even if they're not willing to admit it, a lot of people are scared by what they see transpiring in this world, and they're looking for some sort of hope for the future.