On this account, every belief involves the self-ascription of a property and so, arguably, is an instance of self-consciousness
If you say, "I believe P", it may imply you are conscious that "you believe 'that P'"
But can you believe something that you're not conscious of? Does saying 'I believe P' imply, you're actually conscious of P?
for example, If I say "I believe that "All Cats have 5 legs" could it mean that I am conscious that "All Cats have 5 Legs" or does it mean that I am conscious that I believe "All Cats have 5 Legs". If the later is acceptable, then it is probably valid to say that beliefs, even visibly wrong ones, have self-ascription (self-indicating) property. That is, they indicate I am aware of 'What I believe'.