'Thanks, God' is natural only in the rather flippant context of somebody who, having received a great stroke of good fortune, says these words. S/he is, probably not seriously, addressing God directly.
'Thank God' is an expression of relief. It is used even by atheists. Similar expressions are 'Thank the Lord' and 'Thank Heavens'.
Believers who wish to express their gratitude to God may use a wide variety of expressions, I'll leave it for a believer to suggest some.
Student or Learner