Without any modifications to the hardware, it isn't going to show up in the browser. How I see it, there are two things you can do:
1) You can buy an FMCB memory card online, so that you can enter
wLaunchElf, and that way install
FHDB on the unit, and thus be able to boot homebrew, load games, etc. from the HDD;
or 2), which is probably the best option: Download an HDD image pre-flashed with FHDB already set up (you can use google to find this), and flash it to your HDD with your computer to gain access to wLaunchElf and FHDB.
Be careful to turn the PS2 completely off before you remove the HDD from the PS2, as you might blow a small fuse inside the PS2 otherwise, which is supposed to be a real pain to replace. Also, I've found that the drive needs to be in Cable Select mode when you use it in the PS2, but I don't know if that is true for all HDDs.
Edit: If you go for option number two, it might be a good idea to first create a bootable FMCB memory card and store it somewhere safe, and then reformatting the drive with wLaunchElf before reinstalling FHDB.