As far as I know, Quake 3 does not support any of the Quake 1 binary data formats. Maps (BSP), models (MDL), textures, palettes, container (PAK), .wad, and whatever else there is. Actually Q1 might use .wav audio like Q3? I haven't done any Quake 1 modding so I have limited knowledge on that side of the conversion. I worked on porting a Q1 deathmatch .map file to Q3 BSP once though (never completed or released).
It's possible to convert Q1 level BSPs to human readable text .map files (I don't know what program) and then use GtkRadiant/NetRadiant (q3map2.exe) to make a Q3 level BSP file. Additional work would be needed for textures to work as they need to be TGA or JPG files in
textures/ and/or use Q3 shaders. You will probably need to edit the map entities. Some entity features from Q1 aren't in Q3 (specifically some door feature but I forgot what it was), some items have different names, and some items are too close to wall or floor for them to spawn. If I remember correctly the light entity intensities need to be decreased.
The brush work in Quake 1 maps has many overlaps causing Z-fighting in game, so you will need to manually resize various brushes for the walls/floors. Q1 maps have textures on all sides of brushes instead of marking non-visible sides with
common/calk, so they don't get removed from the Q3 BSP which may cause worse rendering performance.
It's not a simple direct conversion. I'd recommend learning to make Q3 maps, textures, and shaders first.
I don't think you can legally release ports of the official Q1 levels, except the deathmatch maps which were released (without textures) under the GPL. The oa_dm# maps in OpenArena are ports/improved versions of the Q1 deathmatch maps.