No. miners submit their every share submission don't include the pool's bitcoin address, and these submit are just communication between miners and the pool.
I think that is incorrect, if that was the case, i could just mine to a pool, submit shares, get paid, but once i find a block i just put my own address as the coinbase address, which of course can not be done.
The nonce you are hashing is done against many things, one of them is the coinbase transaction which must be in the block header, that transaction ensures the payout goes to the pool address, if you change anything in that transaction the hash you sumbit will be invalid to both the pool and the network as a whole.
The pool keeps all necessary data about block header and transactions for a valid submit to bitcoin network and miners are not given all data by the pool. So only the pool can submit solved block header which includes nonce. Transaction data, sometime, is a huge data; so it won't be given to miners in stratum mode. This answers your curious or doubt.