you load quake 4
drop the in game console down by pressing ctrl+alt+`
then type editor, then the map editor will appear, and you just load the map like a normal program and select all with ctrl+a or whatever in the edit menu
Once you have your map's geometry converted, I wouldn't bother with built-in editor and try GtkRadiant instead, it has Q4 support (gamepack) for years.
Converting could be problematic, however: brush definition is different between the two games, so manual (or semi-automated) conversion is hard, and I'm not sure if Radiant can readily import/export maps in different formats.
You should be able to use QuArK though. Open the map in Q3A mode, save it as a .qrk file (QuArK's native format), change the game tag to Q4 with a text editor, open the map in QuArK in Q4 mode, save again as a .map file (source). I vaguely recall it worked for me, albeit in the opposite direction (getting Q2 version of a Q4 map).
If all fails, try uploading your Q3A map somewhere so we can take a look and perhaps provide you with "raw" Q4 map to work with.
Thank you for your note. It is very much appreciated.
After saving .map to .qrk in QuArK, I can't find where the new .qrk file is saved. Even after all "hidden" settings have been turned off, Windows Explorer and Notepad++ can't still locate it.
Hm, I'm sorry, I might have inadvertently misguided you: apparently, QuArK cannot be forced to output Q4 brush definitions, at least I didn't find a way.
I was about to write a tool for that, but quick googling revealed this little utility which almost did the job: it produced the map with Q4 brushes, albeit not quite correct.
To be able to open the map in GtkRadiant 1.5 in Q4 mode I had to change version from 2 to 3 and fix some syntactic bogosities. I've uploaded it here.
I managed to open the file you uploaded (gimap01_qt4.map) in QuArK. However, when I try to open it with GtkRadiant 1.6.4, so many errors messages appear.
Btw, before this issue gets off target, I will iMsg you just to give you the idea behind this conversion and so forth.