The fact is, it is incorrect to say that it is not biblical to question God. As one of the previous answers said, there's a difference between asking a question and accusing God. The other aspect is that there is a difference between asking God questions with a sincere heart seeking an honest answer and a cynical heart seeking to deride God.
There are many instances in the Bible where God allows individuals to ask Him questions or where He asks them to reason with Him.
“ Present your case,” says the LORD. “ Bring forth your strong reasons,” says the King of Jacob.
Isaiah 41:20-22
"Come now, let us reason together," says the LORD. "Though your sins are like scarlet, they shall be as white as snow; though they are red as crimson, they shall be like wool.
Isaiah 1:17-19
Genesis 32:39
Jacob said, "Please tell me your name."
But he replied, "Why do you ask my name?" Then he blessed him there.
Exodus 3:11 - 14
11 But Moses said to God, "Who am I, that I should go to Pharaoh and bring the Israelites out of Egypt?"
12 And God said, "I will be with you. And this will be the sign to you that it is I who have sent you: When you have brought the people out of Egypt, you [a] will worship God on this mountain."
13 Moses said to God, "Suppose I go to the Israelites and say to them, 'The God of your fathers has sent me to you,' and they ask me, 'What is his name?' Then what shall I tell them?"
14 God said to Moses, "I am who I am . [b] This is what you are to say to the Israelites: 'I AM has sent me to you.' "
As you can see in all these instances, individuals asked God questions and they got answers.
Many people out there would rather see God as some tyrant who does not care and who does not tolerate any form of thought in His children. That is unfortunately not so. The real truth is, God sees beyond the spoken statements or "questions" into the thoughts behind them. He deals with the individual on the basis of those thoughts.
Hebrews 4:12
For the word of God is living and powerful, and sharper than any two-edged sword, piercing even to the division of soul and spirit, and of joints and marrow, and is a discerner of the thoughts and intents of the heart.
I hope this answers your question.