You could declare a process global variable that holds the url.
B4X:
Sub Process_Globals
Dim url As String
End Sub
Sub Button1_Click
url = Url1
StartActivity(WebView) 'with Url1
End Sub
Sub Button2_Click
url = Url2
StartActivity(WebView) 'with Url2
End Sub
And in the WebView activity you can get the url with Main.url
To be able to give you a concrete answer, you should post your project as a zip file (IDE menu Files / Export As Zip) so we could test it in the same conditions as you.