The Bible says God is perfect in knowledge, knows all thoughts, all secrets, sees all and no-one can hide from God.
Talk no more so very proudly;
Let no arrogance come from your mouth,
For the LORD is the God of knowledge;
And by Him actions are weighed. 1 Samuel 2:3
for he views the ends of the earth
and sees everything under the heavens. Job 28:24
Do you know how the clouds are balanced,
Those wondrous works of Him who is perfect in knowledge? Job 37:16
I know that you can do all things;
no plan of yours can be thwarted. Job 42:2
would not God have discovered it,
since he knows the secrets of the heart? Psalm 44:2
For there is not a word on my tongue,
But behold, O LORD, You know it altogether Psalm 139:4,7-8
Where can I go from Your Spirit?
Or where can I flee from Your presence?
8 If I ascend into heaven, You are there;
If I make my bed in hell, behold, You are there.Psalm 139:7-8
Great is our Lord, and mighty in power;
His understanding is infinite.Psalm 147:5
The eyes of the LORD are in every place,
Keeping watch on the evil and the good. Proverbs 15:3
For My eyes are on all their ways; they are not hidden from My face, nor is their iniquity hidden from My eyes. Jeremiah 16:17
Can anyone hide himself in secret places,
So I shall not see him?” says the LORD;
“ Do I not fill heaven and earth?” says the LORD. Jeremiah 23:24
And they prayed and said, “You, O Lord, who know the hearts of all, show which of these two You have chosen Acts 1:24
And there is no creature hidden from His sight, but all things are naked and open to the eyes of Him to whom we must give account. Hebrews 4:13
And by this we know[a] that we are of the truth, and shall assure our hearts before Him. 20 For if our heart condemns us, God is greater than our heart, and knows all things 1 John 3:19-20
There are some epistemological problems with an omniscient god, but I don't know if it's in the scope of this site and it certainly is not written in the bible.