No, but it's probably easiest.
As has been pointed out in the comments, Time Machine requires a drive formatted as Mac OS Extended (HFS+). I believe this requirement can be waived if you're Time Machining to a network drive (Time Machine creates its own HFS+ disk image), but you don't want to buy extra hardware.
You can get read/write Windows drivers for HFS+ (e.g.), but as far as I know they all cost. I also don't know how well your favourite Windows backup utility would cope with an HFS+ drive - if you do go down this route, check that recovery works before you need it.
If you're willing to accept an alternative Mac backup program to Time Machine, you could format the drive as FAT32, but that's an old file system, and has various limits - most prominently, a maximum file size of just 4GB. You could also install NTFS drivers on your Mac (e.g.) and there are free options this time. That said, with both NTFS and FAT32, there is the risk that some Mac files (mainly older ones, admittedly) will have problems because their resource fork gets mislaid in either the backup or recovery.
So, in short, you're probably best using Disk Utility in
/Applications/Utilities to partition your drive into an HFS+ partition for your Mac backups and an NTFS partition for your Windows backups. That way, each operating system can deal with its normal file system, and files in the backup will be on the same type of file system as they are in the original, which means one less possible problem if the worst happens and you need to restore.