When we use the Transfer to Phone number contact block in the Amazon Connect contact flow, we hear a ringing tone while transferring a call to a specific number, and we don't want to play that ringing

"When we use the Transfer to Phone number contact block in the Amazon Connect contact flow, we hear a ringing tone while transferring a call to a specific number," Question - where is this "specific number"? Is it another Connect number or queue in your same instance? If so, you should be able to do a transfer to queue or transfer to flow instead.

Otherwise, if the specific number is not in your Connect instance, I believe the default behavior of transferring calls across the PSTN will include a ringback tone played to the caller. But see also dmacias' answer about sip normalization rules in your other question you asked.