I have an 2 TB external hard drive. It is formatted as NTFS format, supported by ntfs-3g which is installed using Homebrew. There is already about 500 GB data existing on it that I need to keep. I now want to use the rest of the space on it to store my TimeMachine backups. There is only one partition on the hard drive. I do not have other hard drives to shuffle the data. How can I start to use this hard drive to store my Time Machine backups?
Time Machine can only work on a volume formatted as HFS+. So you will need to repartition this drive. A quick Google shows that it is possible to resize an NTFS partition, but this can't be done with Disk Utility on MacOS.
I wouldn't recommend playing with the partitioning on the volume if the 500Gb of stuff isn't also backed up somewhere else.