No the formated drive will work in any system that accepts that type of HDD. But if it has Windows on it it is very unlikely to work as when you install windows you install the drivers for that M'Board and other peripherals.
Without a Floppy or CD ROM your only option is to push out the install across the network or better still as it is easier is to partition the drive and copy the Windows Install Files from a OEM or Retail CD as the Restore CD for the other NB will not have the correct drivers for that NB.
With the complete CD copied to a partition on the HDD you should be able to just then boot up the computer and it should enter the setup but there may be some problems when the thing needs to do a reboot as it would be looking at the primary partition on the HDD and seeing the contents of the Install CD there.
Over all I would try to push out the install over a CAT % Crossover cable if you have a RJ45 socket on this NB, or better still shell out for a working CD ROM and fit it to the NB so you'll have a fighting chance of getting it loaded as there are very few ways other than networking of installing anything at all onto it. If there is no LAN socket you'll need a PCMCIA Network Card and even then it's not going to be an easy job.
If you have a Com port on it you could also try using a Null Modem Cable if they are still available to do the same thing as a network card or socket will do.
Col