How can you tell that a believer is maturing?

I would say a "mature" believer is one who is able to argue their position intelligently, and sees challenging (and often hostile) questions as a chance to look at their own faith more deeply and gives the incentive to study things in more detail, rather than becoming extremely defensive and giving a "God said it, so I believe it, and that settles it" style of answer.

Being an expert and teacher of the word of God is not a concrete sign of spiritual maturity. One can be well verse of the word of God and yet be as carnal as anybody else. A spiritually mature person is someone who has overcome himself by weeding out the worthless and sorrow producing elements of life, such as lust, self-seeking, vanity, self-indulgence, avarice, delusion, etc. and in exchange has transformed with divine characteristics, such as purity, patience, humility, compassion, love, self-sacrifice. The sign of spiritually maturity can not be hidden and it is not hard to detect. An unchanging sweetness of disposition in spite of open hostility is a sign of a self-conquered soul and the mark of a spiritually mature person.

According to the Bible, we can count ourselves mature not merely by our knowledge of the Word, but by our walk in Christ. Ephesians 4 talks about being united in faith with fellow believers, and in our knowledge of Christ. When we're no longer swayed by "every wind of teaching" out there, and living up to what we've already learned can we consider ourselves mature. Other texts to look into: Philippians 3; Colossians 4; Hebrews 5:14; James 1.

No one can measure maturity on his or her religion unless you know the person fully. In order to know how deeply the person believes, you have to know his ways and beliefs.

i don't think knowledge of the bible is the best measure of maturity as a believer. it's more of being able to share with others and to live up to the teachings of God and adapting it in our lives. aside from knowing and understanding His words, having a lot of faith counts most. it is with faith that we surrender ourselves to Him as a whole and we see our actions as a way of thanking, glorifying, and giving back to Him all the good He has given us.