No, you cannot. Your Time Machine backup isn't a real clone of your hard drive — it's a time-based collection of all the files on your hard drive.
It's a little intensive to do this process, so proceed with caution if you follow the steps below. Read through them — if you feel uncomfortable with any of the steps, feel free to read up on anything you're not comfortable with or hire a friend or professional who is more familiar with this kind of stuff.
If you know that the 500GB hard drive will actually fit in your computer (i.e. your current internal hard drive and the one you want to replace it with are both 2.5"), then you can put it in. If you have another external hard drive you can use temporarily, you can switch them out without losing your backups; if you only have the 500GB external hard drive, you can do still do it, but you'll lose your old backups.
I would do the following steps:
- Create a bootable MacOS USB drive
- Shut down your computer and replace the internal hard drive with the 500GB hard drive
- Wipe the 500GB hard disk and reinstall MacOS using the MacOS USB drive
- On first boot during the initial setup process, select the option to restore from another computer hard drive
- After that is done, you are free to wipe the 250GB hard drive and use it for your future Time Machine backups