The answer depends on how (and if) you chose to migrate content to your new Mac. If you chose to use Migration Assistant when offered the opportunity, and you chose the previous Mac's Time Machine backup for the source of that content, then TM will pick up where it left off and will continue as it always has. Meaning, older and "expired" backups will fall off as they age, as they always have, thereby releasing space for newer ones.
If you declined that opportunity, and either chose to migrate content from the older Mac directly or elected to set up the new Mac as a completely new device, then the older one's Time Machine backup will reside on that backup disk forever, unless of course you decide to erase it and start a new set of backups for that new Mac.
If you do not erase that backup disk, and it has been in service long enough to have retained a great number of backups, it may have insufficient capacity to start even one backup for the new Mac.
You can always choose to erase the new Mac and start over, if you wish to do things differently.