If you don't believe God exists then you believe God doesn't exist. This is what Wikipedia is saying above. God either exists or doesn't.Atheism is, in a broad sense, the rejection of belief in the existence of deities. In a narrower sense, atheism is specifically the position that there are no deities. Most inclusively, atheism is simply the absence of belief that any deities exist (Wikipedia)
To an atheist the proposition "God exists" is false
To an atheist the proposition "God doesn't exist" is true, therefore he believes, or thinks, or asserts, or has the position, or ... whatever that God doesn't exist.