If the icon on your laptop is just a Web shortcut that takes your browser to a page on the Wi-Fi router configuration pages, then you may be able to browse to the same web location using the Hotspot Browser on your blackberry. Then enter the user name/password on the Blackberry as you would on your laptop.
If this sounds correct, then what I suggest you do is this:
1) Go through the steps for connecting with your laptop, but don't enter the username/password.
2) Copy the URL (starts with http://)
from the field at the top of the browser and paste it into an email that you can send to your Blackberry.
3) While waiting for the email to arrive on your Blackberry, set the default browser to Hotspot: Start your browser, bring up the menu and select Options. Then General Properties. On that page set the Default Browser to Hotspot Browser.
4) When the email with the URL comes in, open the email and select the link. It should take you to the same configuration page as you see on your lap top. If we are lucky the Blackberry will display it close enough to the way your laptop does for you to enter the username/password and get it to work.
5) If this has all worked, save the link from the email as a bookmark and you should be all set.
Otherwise come back and tell me where I've gone wrong. We may be able to try something else.