Peace is one of the fruits of the Spirit. It's what a mature believer should have. And not just peace.
(Galatians 5:22-23 NIV) But the fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faithfulness, {23} gentleness and self-control. Against such things there is no law.
Thank you for your comment, Barncat. As a sidenote to anyone who thinks they need to be a 'mature' believer to know God's peace, I wouldn't put myself in the 'mature' category.. I wouldn't have even called myself a Christian at first.
I see it this way, God's peace comes when God is put center-stage in our life. It's the presence of the Holy Spirit. The fruit of the Holy Spirit comes naturally as we walk more and more with the Holy Spirit. It's not our peace/love/patience/kindness etc but Jesus'. Then I just ask Jesus for His peace, wisdom, patience, self control, gentleness etc. which, thankfully, he gives. I can't do anything for the fruit of the Spirit. It's always unbelievable how it comes suddenly.
Before I became born again, self-development/growth/maturity etc was like an idol to me. I condemned myself for being immature/broken/imperfect etc. All those burdens were released when I repented and saw God's peace, love and joy is not dependent on how I am/what I do. It's His gift.