Yes, bittrex support for bitcoin gold, for complete source of information please visit this link https://support.bittrex.com/hc/en-us/articles/115002320451-Statement-on-Bitcoin-Gold-BTG-
Right now Bitcoin Gold has just implemented a solution called SIGHASH_FORK_ID replay protection, the update continues. It is an effective two-way replay protection mechanism that enforces a new algorithm to calculate the hash of a transaction so that all new Bitcoin transactions will be invalid on the Bitcoin Gold blockchain and vice versa. The update for replay protection can be checked on github here https://github.com/BTCGPU/BTCGPU/pull/109